2014
2014
-
Paper 1, Section I, D
2014 commentShow that every sequence of real numbers contains a monotone subsequence.
-
Paper 1, Section I,
2014 commentFind the radius of convergence of the following power series: (i) ; (ii) .
-
Paper 1, Section II, D
2014 comment(a) Show that for all ,
stating carefully what properties of sin you are using.
Show that the series converges absolutely for all .
(b) Let be a decreasing sequence of positive real numbers tending to zero. Show that for not a multiple of , the series
converges.
Hence, or otherwise, show that converges for all .
-
Paper 1, Section II, E
2014 comment(i) State the Mean Value Theorem. Use it to show that if is a differentiable function whose derivative is identically zero, then is constant.
(ii) Let be a function and a real number such that for all ,
Show that is continuous. Show moreover that if then is constant.
(iii) Let be continuous, and differentiable on . Assume also that the right derivative of at exists; that is, the limit
exists. Show that for any there exists satisfying
[You should not assume that is continuous.]
-
Paper 1, Section II, E
2014 comment(i) Prove Taylor's Theorem for a function differentiable times, in the following form: for every there exists with such that
[You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]
(ii) The function is twice differentiable, and satisfies the differential equation with . Show that is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to at every point. Hence or otherwise show that for any , the series
converges to .
-
Paper 1, Section II, F
2014 commentDefine what it means for a function to be (Riemann) integrable. Prove that is integrable whenever it is
(a) continuous,
(b) monotonic.
Let be an enumeration of all rational numbers in . Define a function by ,
where
Show that has a point of discontinuity in every interval .
Is integrable? [Justify your answer.]
-
Paper 2, Section I, B
2014 commentThe following equation arises in the theory of elastic beams:
where is a real valued function.
By using the change of variables
find the general solution of the above equation.
-
Paper 2, Section I, B
2014 commentConsider the ordinary differential equation
State an equation to be satisfied by and that ensures that equation is exact. In this case, express the general solution of equation in terms of a function which should be defined in terms of and .
Consider the equation
satisfying the boundary condition . Find an explicit relation between and .
-
Paper 2, Section II, B
2014 commentUse the transformation
where is a constant, to map the Ricatti equation
to a linear equation.
Using the above result, as well as the change of variables , solve the boundary value problem
where is a positive constant. What is the value of for which the solution is singular?
-
Paper 2, Section II, B
2014 commentThe so-called "shallow water theory" is characterised by the equations
where denotes the gravitational constant, the constant denotes the undisturbed depth of the water, denotes the speed in the -direction, and denotes the elevation of the water.
(i) Assuming that and and their gradients are small in some appropriate dimensional considerations, show that satisfies the wave equation
where the constant should be determined in terms of and .
(ii) Using the change of variables
show that the general solution of satisfying the initial conditions
is given by
where
Simplify the above to find in terms of and .
(iii) Find in the particular case that
where denotes the Heaviside step function.
Describe in words this solution.
-
Paper 2, Section II, B
2014 comment(a) Let be a solution of the equation
Assuming that the second linearly independent solution takes the form , derive an ordinary differential equation for .
(b) Consider the equation
By inspection or otherwise, find an explicit solution of this equation. Use the result in (a) to find the solution satisfying the conditions
-
Paper 2, Section II, B
2014 commentConsider the damped pendulum equation
where is a positive constant. The energy , which is the sum of the kinetic energy and the potential energy, is defined by
(i) Verify that is a decreasing function.
(ii) Assuming that is sufficiently small, so that terms of order can be neglected, find an approximation for the general solution of in terms of two arbitrary constants. Discuss the dependence of this approximate solution on .
(iii) By rewriting as a system of equations for and , find all stationary points of and discuss their nature for all , except .
(iv) Draw the phase plane curves for the particular case .
-
Paper 4, Section I,
2014 commentA particle of mass has charge and moves in a constant magnetic field B. Show that the particle's path describes a helix. In which direction is the axis of the helix, and what is the particle's rotational angular frequency about that axis?
-
Paper 4, Section I,
2014 commentWhat is a 4-vector? Define the inner product of two 4-vectors and give the meanings of the terms timelike, null and spacelike. How do the four components of a 4-vector change under a Lorentz transformation of speed ? [Without loss of generality, you may take the velocity of the transformation to be along the positive -axis.]
Show that a 4-vector that is timelike in one frame of reference is also timelike in a second frame of reference related by a Lorentz transformation. [Again, you may without loss of generality take the velocity of the transformation to be along the positive -axis.]
Show that any null 4-vector may be written in the form where is real and is a unit 3-vector. Given any two null 4-vectors that are future-pointing, that is, which have positive time-components, show that their sum is either null or timelike.
-
Paper 4, Section II, C
2014 commentA rocket of mass , which includes the mass of its fuel and everything on board, moves through free space in a straight line at speed . When its engines are operational, they burn fuel at a constant mass rate and eject the waste gases behind the rocket at a constant speed relative to the rocket. Obtain the rocket equation
The rocket is initially at rest in a cloud of space dust which is also at rest. The engines are started and, as the rocket travels through the cloud, it collects dust which it stores on board for research purposes. The mass of dust collected in a time is given by , where is the distance travelled in that time and is a constant. Obtain the new equations
By eliminating , or otherwise, obtain the relationship
where is the initial mass of the rocket and .
If , show that the fuel will be exhausted before the speed of the rocket can reach . Comment on the case when , giving a physical interpretation of your answer.
-
Paper 4, Section II, C
2014 commentA reference frame rotates with constant angular velocity relative to an inertial frame that has the same origin as . A particle of mass at position vector is subject to a force . Derive the equation of motion for the particle in .
A marble moves on a smooth plane which is inclined at an angle to the horizontal. The whole plane rotates at constant angular speed about a vertical axis through a point fixed in the plane. Coordinates are defined with respect to axes fixed in the plane: horizontal and up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys
By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.
[You may assume that the marble never leaves the plane.]
-
Paper 4, Section II, C
2014 commentA thin flat disc of radius has density (mass per unit area) where are plane polar coordinates on the disc and is a constant. The disc is free to rotate about a light, thin rod that is rigidly fixed in space, passing through the centre of the disc orthogonal to it. Find the moment of inertia of the disc about the rod.
The section of the disc lying in is cut out and removed. Starting from rest, a constant torque is applied to the remaining part of the disc until its angular speed about the axis reaches . Show that this takes a time
After this time, no further torque is applied and the partial disc continues to rotate at constant angular speed . Given that the total mass of the partial disc is , where is a constant that you need not determine, find the position of the centre of mass, and hence its acceleration. From where does the force required to produce this acceleration arise?
-
Paper 4, Section II, C
2014 commentDefine the 4-momentum of a particle and describe briefly the principle of conservation of 4-momentum.
A photon of angular frequency is absorbed by a particle of rest mass that is stationary in the laboratory frame of reference. The particle then splits into two equal particles, each of rest mass .
Find the maximum possible value of as a function of . Verify that as , this maximum value tends to . For general , show that when the maximum value of is achieved, the resulting particles are each travelling at speed in the laboratory frame.
-
Paper 3, Section I, D
2014 commentLet be the rational numbers, with addition as the group operation. Let be non-zero elements of , and let be the subgroup they generate. Show that is isomorphic to .
Find non-zero elements which generate a subgroup that is not isomorphic to .
-
Paper 3, Section I, D
2014 commentLet be a group, and suppose the centre of is trivial. If divides , show that has a non-trivial conjugacy class whose order is prime to .
-
Paper 3, Section II, D
2014 commentLet be the group of permutations of , and suppose is even, .
Let , and .
(i) Compute the centraliser of , and the orders of the centraliser of and of the centraliser of .
(ii) Now let . Let be the group of all symmetries of the cube, and the set of faces of the cube. Show that the action of on makes isomorphic to the centraliser of in . [Hint: Show that permutes the faces of the cube according to .]
Show that is also isomorphic to the centraliser of in .
-
Paper 3, Section II, D
2014 commentLet be a prime number. Let be a group such that every non-identity element of has order .
(i) Show that if is finite, then for some . [You must prove any theorems that you use.]
(ii) Show that if , and , then .
Hence show that if is abelian, and is finite, then .
(iii) Let be the set of all matrices of the form
where and is the field of integers modulo . Show that every nonidentity element of has order if and only if . [You may assume that is a subgroup of the group of all invertible matrices.]
-
Paper 3, Section II, D
2014 commentLet be a prime number, and , the group of invertible matrices with entries in the field of integers modulo .
The group acts on by Möbius transformations,
(i) Show that given any distinct there exists such that , and . How many such are there?
(ii) acts on by . Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.
-
Paper 3, Section II, D
2014 comment(a) Let be a group, and a subgroup of . Define what it means for to be normal in , and show that if is normal then naturally has the structure of a group.
(b) For each of (i)-(iii) below, give an example of a non-trivial finite group and non-trivial normal subgroup satisfying the stated properties.
(i) .
(ii) There is no group homomorphism such that the composite is the identity.
(iii) There is a group homomorphism such that the composite is the identity, but the map
is not a group homomorphism.
Show also that for any satisfying (iii), this map is always a bijection.
-
Paper 4, Section I, E
2014 commentUse Euclid's algorithm to determine , the greatest common divisor of 203 and 147 , and to express it in the form for integers . Hence find all solutions in integers of the equation .
How many integers are there with and
-
Paper 4, Section I,
2014 commentDefine the binomial coefficients , for integers satisfying . Prove directly from your definition that if then
and that for every and ,
-
Paper 4, Section II, E
2014 commentWhat does it mean to say that the sequence of real numbers converges to the limit What does it mean to say that the series converges to ?
Let and be convergent series of positive real numbers. Suppose that is a sequence of positive real numbers such that for every , either or . Show that is convergent.
Show that is convergent, and that is divergent if .
Let be a sequence of positive real numbers such that is convergent. Show that is convergent. Determine (with proof or counterexample) whether or not the converse statement holds.
-
Paper 4, Section II, E
2014 comment(i) State and prove the Fermat-Euler Theorem.
(ii) Let be an odd prime number, and an integer coprime to . Show that , and that if the congruence has a solution then .
(iii) By arranging the residue classes coprime to into pairs with , or otherwise, show that if the congruence has no solution then
(iv) Show that .
-
Paper 4, Section II, E
2014 comment(i) What does it mean to say that a set is countable? Show directly that the set of sequences , with for all , is uncountable.
(ii) Let be any subset of . Show that there exists a bijection such that (the set of even natural numbers) if and only if both and its complement are infinite.
(iii) Let be the binary expansion of . Let be the set of all sequences with such that for infinitely many . Let be the set of all such that for infinitely many . Show that is uncountable.
-
Paper 4, Section II, E
2014 comment(i) State and prove the Inclusion-Exclusion Principle.
(ii) Let be an integer. Denote by the integers modulo . Let be the set of all functions such that for every . Show that
-
Paper 2, Section I, F
2014 commentConsider a particle situated at the origin of . At successive times a direction is chosen independently by picking an angle uniformly at random in the interval , and the particle then moves an Euclidean unit length in this direction. Find the expected squared Euclidean distance of the particle from the origin after such movements.
-
Paper 2, Section I, F
2014 commentConsider independent discrete random variables and assume exists for all .
Show that
If the are also positive, show that
-
Paper 2, Section II, 9F
2014 commentState the axioms of probability.
State and prove Boole's inequality.
Suppose you toss a sequence of coins, the -th of which comes up heads with probability , where . Calculate the probability of the event that infinitely many heads occur.
Suppose you repeatedly and independently roll a pair of fair dice and each time record the sum of the dice. What is the probability that an outcome of 5 appears before an outcome of 7 ? Justify your answer.
-
Paper 2, Section II, F
2014 commentDefine what it means for a random variable to have a Poisson distribution, and find its moment generating function.
Suppose are independent Poisson random variables with parameters . Find the distribution of .
If are independent Poisson random variables with parameter , find the distribution of . Hence or otherwise, find the limit of the real sequence
[Standard results may be used without proof provided they are clearly stated.]
-
Paper 2, Section II, F
2014 commentFor any function and random variables , the "tower property" of conditional expectations is
Provide a proof of this property when both are discrete.
Let be a sequence of independent uniform -random variables. For find the expected number of 's needed such that their sum exceeds , that is, find where
[Hint: Write
-
Paper 2, Section II, F
2014 commentGive the definition of an exponential random variable with parameter . Show that is memoryless.
Now let be independent exponential random variables, each with parameter . Find the probability density function of the random variable and the probability .
Suppose the random variables are independent and each has probability density function given by
Find the probability density function of [You may use standard results without proof provided they are clearly stated.]
-
Paper 3, Section I, A
2014 comment(a) For and , show that
(b) Use index notation and your result in (a), or otherwise, to compute
(i) , and
(ii) for .
(c) Show that for each there is, up to an arbitrary constant, just one vector field of the form such that everywhere on , and determine .
-
Paper 3, Section I, A
2014 commentLet be a vector field defined everywhere on the domain .
(a) Suppose that has a potential such that for . Show that
for any smooth path from a to in . Show further that necessarily on .
(b) State a condition for which ensures that implies is pathindependent.
(c) Compute the line integral for the vector field
where denotes the anti-clockwise path around the unit circle in the -plane. Compute and comment on your result in the light of (b).
-
Paper 3, Section II, A
2014 commentThe surface in is given by .
(a) Show that the vector field
is tangent to the surface everywhere.
(b) Show that the surface integral is a constant independent of for any surface which is a subset of , and determine this constant.
(c) The volume in is bounded by the surface and by the cylinder . Sketch and compute the volume integral
directly by integrating over .
(d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral , where is the portion of lying in .
-
Paper 3, Section II, A
2014 comment(a) State Stokes' Theorem for a surface with boundary .
(b) Let be the surface in given by where . Sketch the surface and find the surface element with respect to the Cartesian coordinates and .
(c) Compute for the vector field
and verify Stokes' Theorem for on the surface .
-
Paper 3, Section II, 11A
2014 comment(i) Starting with Poisson's equation in ,
derive Gauss' flux theorem
for and for any volume .
(ii) Let
Show that if is the sphere , and that if bounds a volume that does not contain the origin.
(iii) Show that the electric field defined by
satisfies
where is a surface bounding a closed volume and , and where the electric charge and permittivity of free space are constants. This is Gauss' law for a point electric charge.
(iv) Assume that is spherically symmetric around the origin, i.e., it is a function only of . Assume that is also spherically symmetric. Show that depends only on the values of inside the sphere with radius but not on the values of outside this sphere.
-
Paper 3, Section II, A
2014 comment(a) Show that any rank 2 tensor can be written uniquely as a sum of two rank 2 tensors and where is symmetric and is antisymmetric.
(b) Assume that the rank 2 tensor is invariant under any rotation about the -axis, as well as under a rotation of angle about any axis in the -plane through the origin.
(i) Show that there exist such that can be written as
(ii) Is there some proper subgroup of the rotations specified above for which the result still holds if the invariance of is restricted to this subgroup? If so, specify the smallest such subgroup.
(c) The array of numbers is such that is a vector for any symmetric matrix .
(i) By writing as a sum of and with and , show that is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]
(ii) Does necessarily have to be a tensor? Justify your answer.
-
Paper 1, Section I, 1B
2014 comment(a) Let
(i) Compute .
(ii) Find all complex numbers such that .
(b) Find all the solutions of the equation
(c) Let . Show that the equation of any line, and of any circle, may be written respectively as
for some complex and real .
-
Paper 1, Section I, 2A
2014 comment(a) What is meant by an eigenvector and the corresponding eigenvalue of a matrix ?
(b) Let be the matrix
Find the eigenvalues and the corresponding eigenspaces of and determine whether or not is diagonalisable.
-
Paper 1, Section II, 5B
2014 comment(i) For vectors , show that
Show that the plane and the line , where , intersect at the point
and only at that point. What happens if ?
(ii) Explain why the distance between the planes and is , where is a unit vector.
(iii) Find the shortest distance between the lines and where . [You may wish to consider two appropriately chosen planes and use the result of part (ii).]
-
Paper 1, Section II, A
2014 commentLet be a real symmetric matrix.
(i) Show that all eigenvalues of are real, and that the eigenvectors of with respect to different eigenvalues are orthogonal. Assuming that any real symmetric matrix can be diagonalised, show that there exists an orthonormal basis of eigenvectors of .
(ii) Consider the linear system
Show that this system has a solution if and only if for every vector in the kernel of . Let be such a solution. Given an eigenvector of with non-zero eigenvalue, determine the component of in the direction of this eigenvector. Use this result to find the general solution of the linear system, in the form
-
Paper 1, Section II,
2014 commentLet be the linear map
where and are real constants. Write down the matrix of with respect to the standard basis of and show that .
Let be the invertible map
and define a linear map by . Find the image of each of the standard basis vectors of under both and . Hence, or otherwise, find the matrix of with respect to the standard basis of and verify that .
-
Paper 3, Section I, F
2014 commentLet be an open set and let be a differentiable function on such that for some constant and all , where denotes the operator norm of the linear map . Let be a straight-line segment contained in . Prove that , where denotes the Euclidean norm on .
Prove that if is an open ball and for each , then is constant on .
-
Paper 4, Section I, F
2014 commentDefine a contraction mapping and state the contraction mapping theorem.
Let be the space of continuous real-valued functions on endowed with the uniform norm. Show that the map defined by
is not a contraction mapping, but that is.
-
Paper 2, Section I, F
2014 commentDefine what is meant by a uniformly continuous function on a set .
If and are uniformly continuous functions on , is the (pointwise) product necessarily uniformly continuous on ?
Is a uniformly continuous function on necessarily bounded?
Is uniformly continuous on
Justify your answers.
-
Paper 1, Section II, F
2014 commentDefine what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms on are Lipschitz equivalent and , then is closed in one norm if and only if is closed in the other norm.
Show that if is finite-dimensional, then any two norms on are Lipschitz equivalent.
Show that is a norm on the space of continuous realvalued functions on . Is the set closed in the norm ?
Determine whether or not the norm is Lipschitz equivalent to the uniform on .
[You may assume the Bolzano-Weierstrass theorem for sequences in .]
-
Paper 4, Section II, F
2014 commentLet be an open set. Define what it means for a function to be differentiable at a point .
Prove that if the partial derivatives and exist on and are continuous at , then is differentiable at .
If is differentiable on must be continuous at Give a proof or counterexample as appropriate.
The function is defined by
Determine all the points at which is differentiable.
-
Paper 3, Section II, F
2014 commentLet , be continuous functions on an open interval . Prove that if the sequence converges to uniformly on then the function is continuous on .
If instead is only known to converge pointwise to and is continuous, must be uniformly convergent? Justify your answer.
Suppose that a function has a continuous derivative on and let
Stating clearly any standard results that you require, show that the functions converge uniformly to on each interval .
-
Paper 2, Section II,
2014 commentLet be subsets of and define . For each of the following statements give a proof or a counterexample (with justification) as appropriate.
(i) If each of is bounded and closed, then is bounded and closed.
(ii) If is bounded and closed and is closed, then is closed.
(iii) If are both closed, then is closed.
(iv) If is open and is closed, then is open.
[The Bolzano-Weierstrass theorem in may be assumed without proof.]
-
Paper 4, Section I, G
2014 commentLet be an entire function. State Cauchy's Integral Formula, relating the th derivative of at a point with the values of on a circle around .
State Liouville's Theorem, and deduce it from Cauchy's Integral Formula.
Let be an entire function, and suppose that for some we have that for all . Prove that is a polynomial.
-
Paper 1, Section II, B
2014 commentBy choice of a suitable contour show that for
Hence evaluate
using the substitution .
-
Paper 2, Section II, B
2014 commentBy considering a rectangular contour, show that for we have
Hence evaluate
-
Paper 3, Section II, G
2014 commentState the Residue Theorem precisely.
Let be a star-domain, and let be a closed path in . Suppose that is a holomorphic function on , having no zeros on . Let be the number of zeros of inside , counted with multiplicity (i.e. order of zero and winding number). Show that
[The Residue Theorem may be used without proof.]
Now suppose that is another holomorphic function on , also having no zeros on and with on . Explain why, for any , the expression
is well-defined. By considering the behaviour of the function as varies, deduce Rouché's Theorem.
For each , let be the polynomial . Show that, as tends to infinity, the smallest modulus of the roots of also tends to infinity.
[You may assume any results on convergence of power series, provided that they are stated clearly.]
-
Paper 1, Section I, B
2014 commentLet be an analytic/holomorphic function defined on an open set , and let be a point such that . Show that the transformation preserves the angle between smooth curves intersecting at . Find such a transformation that maps the second quadrant of the unit disc (i.e. to the region in the first quadrant of the complex plane where (i.e. the region in the first quadrant outside the unit circle).
-
Paper 3, Section I, B
2014 commentFind the most general cubic form
which satisfies Laplace's equation, where and are all real. Hence find an analytic function which has such a as its real part.
-
Paper 4, Section II, B
2014 commentFind the Laplace transforms of for a positive integer and where and is the Heaviside step function.
Consider a semi-infinite string which is initially at rest and is fixed at one end. The string can support wave-like motions, and for it is allowed to fall under gravity. Therefore the deflection from its initial location satisfies
with
where is a constant. Use Laplace transforms to find .
[The convolution theorem for Laplace transforms may be quoted without proof.]
-
Paper 2, Section I, A
2014 commentStarting from Maxwell's equations, deduce that
for a moving circuit , where is the flux of through the circuit and where the electromotive force is defined to be
where denotes the velocity of a point on .
[Hint: Consider the closed surface consisting of the surface bounded by , the surface bounded by and the surface stretching from to . Show that the flux of through is .]
-
Paper 4, Section I, A
2014 commentA continuous wire of resistance is wound around a very long right circular cylinder of radius , and length (long enough so that end effects can be ignored). There are turns of wire per unit length, wound in a spiral of very small pitch. Initially, the magnetic field is .
Both ends of the coil are attached to a battery of electromotance at , which induces a current . Use Ampère's law to derive inside and outside the cylinder when the displacement current may be neglected. Write the self-inductance of the coil in terms of the quantities given above. Using Ohm's law and Faraday's law of induction, find explicitly in terms of and .
-
Paper 1, Section II, A
2014 commentThe region is occupied by an ideal earthed conductor and a point charge with mass is held above it at .
(i) What are the boundary conditions satisfied by the electric field on the surface of the conductor?
(ii) Consider now a system without the conductor mentioned above. A point charge with mass is held at , and one of charge is held at . Show that the boundary condition on at is identical to the answer to (i). Explain why this represents the electric field due to the charge at under the influence of the conducting boundary.
(iii) The original point charge in (i) is released with zero initial velocity. Find the time taken for the point charge to reach the plane (ignoring gravity).
[You may assume that the force on the point charge is equal to , where is the position vector of the charge, and is time.]
-
Paper 3, Section II, A
2014 comment(i) Consider charges at and at . Write down the electric potential.
(ii) Take . A quadrupole is defined in the limit that such that tends to a constant . Find the quadrupole's potential, showing that it is of the form
where . Determine the constants and .
(iii) The quadrupole is fixed at the origin. At time a particle of charge has the same sign as and mass is at travelling with velocity , where
Neglecting gravity, find the time taken for the particle to reach the quadrupole in terms of , given that the force on the particle is equal to .
-
Paper 2, Section II, A
2014 commentWhat is the relationship between the electric field and the charge per unit area on the surface of a perfect conductor?
Consider a charge distribution distributed with potential over a finite volume within which there is a set of perfect conductors with charges , each at a potential (normalised such that the potential at infinity is zero). Using Maxwell's equations and the divergence theorem, derive a relationship between the electrostatic energy and a volume integral of an explicit function of the electric field , where
Consider concentric perfectly conducting spherical shells. Shell has radius (where ) and charge for , and charge for . Show that
and determine the constant of proportionality.
-
Paper 1, Section I, B
2014 commentConstant density viscous fluid with dynamic viscosity flows in a two-dimensional horizontal channel of depth . There is a constant pressure gradient in the horizontal -direction. The upper horizontal boundary at is driven at constant horizontal speed , with the lower boundary being held at rest. Show that the steady fluid velocity in the -direction is
Show that it is possible to have at some point in the flow for sufficiently large pressure gradient. Derive a relationship between and so that there is no net volume flux along the channel. For the flow with no net volume flux, sketch the velocity profile.
-
Paper 2, Section I, B
2014 commentConsider the steady two-dimensional fluid velocity field
where and . Show that the fluid is incompressible. The streamfunction is defined by , where . Show that is given by
Hence show that the streamlines are defined by
for a constant. For each of the three cases below, sketch the streamlines and briefly describe the flow. (i) , (ii) , (iii) .
-
Paper 1, Section II, B
2014 commentConsider the purely two-dimensional steady flow of an inviscid incompressible constant density fluid in the absence of body forces. For velocity , the vorticity is . Show that
where is the pressure and is the fluid density. Hence show that, if is a constant in both space and time,
where is a constant and is the streamfunction. Here, is defined by , where .
Fluid in the annular region has constant (in both space and time) vorticity . The streamlines are concentric circles, with the fluid speed zero on and on . Calculate the velocity field, and hence show that
Deduce that the pressure difference between the outer and inner edges of the annular region is
[Hint: Note that in cylindrical polar coordinates , the curl of a vector field is
-
Paper 4, Section II, B
2014 commentConsider a layer of fluid of constant density and equilibrium depth in a rotating frame of reference, rotating at constant angular velocity about the vertical -axis. The equations of motion are
where is the fluid pressure, and are the fluid velocities in the -direction and direction respectively, , and is the constant acceleration due to gravity. You may also assume that the horizontal extent of the layer is sufficiently large so that the layer may be considered to be shallow, such that vertical velocities may be neglected.
By considering mass conservation, show that the depth of the layer satisfies
Now assume that , where . Show that the (linearised) potential vorticity , defined by
and is the unit vector in the vertical -direction, is a constant in time, i.e. .
When everywhere, establish that the surface perturbation satisfies
and show that this equation has wave-like solutions when and are related through a dispersion relation to be determined. Show that, to leading order, the trajectories of fluid particles for these waves are ellipses. Assuming that , and , sketch the fluid velocity when for .
-
Paper 3, Section II, B
2014 commentA bubble of gas occupies the spherical region , and an incompressible irrotational liquid of constant density occupies the outer region , such that as the liquid is at rest with constant pressure . Briefly explain why it is appropriate to use a velocity potential to describe the liquid velocity u.
By applying continuity of velocity across the gas-liquid interface, show that the liquid pressure (for ) satisfies
Show that the excess pressure at the bubble surface is
and hence that
The pressure inside the gas bubble satisfies the equation of state
where is a constant, and is the bubble volume. At time the bubble is at rest with radius . If the bubble then expands and comes to rest at , determine the required gas pressure at in terms of .
[You may assume that there is contact between liquid and gas for all time, that all motion is spherically symmetric about the origin , and that there is no body force. You may also assume Bernoulli's integral of the equation of motion to determine the liquid pressure
where is the velocity potential.]
-
Paper 1, Section I, F
2014 commentDetermine the second fundamental form of a surface in defined by the parametrisation
for , with some fixed . Show that the Gaussian curvature of this surface takes both positive and negative values.
-
Paper 3, Section I, F
2014 commentLet be an isometry , where is an matrix and . What are the possible values of ?
Let denote the identity matrix. Show that if and , but , then has a fixed point. Must have a fixed point if and , but Justify your answer.
-
Paper 3, Section II, F
2014 commentLet be a decomposition of the two-dimensional sphere into polygonal domains, with every polygon having at least three edges. Let , and denote the numbers of vertices, edges and faces of , respectively. State Euler's formula. Prove that .
Suppose that at least three edges meet at every vertex of . Let be the number of faces of that have exactly edges and let be the number of vertices at which exactly edges meet . Is it possible for to have ? Justify your answer.
By expressing in terms of the , or otherwise, show that has at least four faces that are triangles, quadrilaterals and/or pentagons.
-
Paper 2, Section II, F
2014 commentLet be the upper half-plane with a hyperbolic metric . Prove that every hyperbolic circle in is also a Euclidean circle. Is the centre of as a hyperbolic circle always the same point as the centre of as a Euclidean circle? Give a proof or counterexample as appropriate.
Let and be two hyperbolic triangles and denote the hyperbolic lengths of their sides by and , respectively. Show that if and , then there is a hyperbolic isometry taking to . Is there always such an isometry if instead the triangles have one angle the same and Justify your answer.
[Standard results on hyperbolic isometries may be assumed, provided they are clearly stated.]
-
Paper 4, Section II, F
2014 commentDefine an embedded parametrised surface in . What is the Riemannian metric induced by a parametrisation? State, in terms of the Riemannian metric, the equations defining a geodesic curve , assuming that is parametrised by arc-length.
Let be a conical surface
Using an appropriate smooth parametrisation, or otherwise, prove that is locally isometric to the Euclidean plane. Show that any two points on can be joined by a geodesic. Is this geodesic always unique (up to a reparametrisation)? Justify your answer.
[The expression for the Euclidean metric in polar coordinates on may be used without proof.]
-
Paper 3, Section I, E
2014 commentState and prove Hilbert's Basis Theorem.
-
Paper 4, Section I, E
2014 commentLet be the abelian group generated by elements and subject to the relations: and . Express as a product of cyclic groups. Hence determine the number of elements of of order 3 .
-
Paper 2, Section I,
2014 commentList the conjugacy classes of and determine their sizes. Hence prove that is simple.
-
Paper 1, Section II, E
2014 commentLet be a finite group and a prime divisor of the order of . Give the definition of a Sylow -subgroup of , and state Sylow's theorems.
Let and be distinct primes. Prove that a group of order is not simple.
Let be a finite group, a normal subgroup of and a Sylow -subgroup of H. Let denote the normaliser of in . Prove that if then there exist and such that .
-
Paper 4, Section II, E
2014 comment(a) Consider the four following types of rings: Principal Ideal Domains, Integral Domains, Fields, and Unique Factorisation Domains. Arrange them in the form (where means if a ring is of type then it is of type )
Prove that these implications hold. [You may assume that irreducibles in a Principal Ideal Domain are prime.] Provide examples, with brief justification, to show that these implications cannot be reversed.
(b) Let be a ring with ideals and satisfying . Define to be the set . Prove that is an ideal of . If and are principal, prove that is principal.
-
Paper 3, Section II, E
2014 commentLet be a ring, an -module and a subset of . Define what it means to say spans . Define what it means to say is an independent set.
We say is a basis for if spans and is an independent set. Prove that the following two statements are equivalent.
-
is a basis for .
-
Every element of is uniquely expressible in the form for some .
We say generates freely if spans and any map , where is an -module, can be extended to an -module homomorphism . Prove that generates freely if and only if is a basis for .
Let be an -module. Are the following statements true or false? Give reasons.
(i) If spans then necessarily contains an independent spanning set for .
(ii) If is an independent subset of then can always be extended to a basis for .
-
-
Paper 2, Section II, 11E
2014 commentProve that every finite integral domain is a field.
Let be a field and an irreducible polynomial in the polynomial ring . Prove that is a field, where denotes the ideal generated by .
Hence construct a field of 4 elements, and write down its multiplication table.
Construct a field of order 9 .
-
Paper 4, Section I, G
2014 commentLet denote the vector space of all real polynomials of degree at most 2 . Show that
defines an inner product on .
Find an orthonormal basis for .
-
Paper 2, Section I, G
2014 commentState and prove the Rank-Nullity Theorem.
Let be a linear map from to . What are the possible dimensions of the kernel of ? Justify your answer.
-
Paper 1, Section I, G
2014 commentState and prove the Steinitz Exchange Lemma. Use it to prove that, in a finitedimensional vector space: any two bases have the same size, and every linearly independent set extends to a basis.
Let be the standard basis for . Is a basis for Is a basis for Justify your answers.
-
Paper 1, Section II, G
2014 commentLet be an -dimensional real vector space, and let be an endomorphism of . We say that acts on a subspace if .
(i) For any , show that acts on the linear span of .
(ii) If spans , show directly (i.e. without using the CayleyHamilton Theorem) that satisfies its own characteristic equation.
(iii) Suppose that acts on a subspace with and . Let be a basis for , and extend to a basis for . Describe the matrix of with respect to this basis.
(iv) Using (i), (ii) and (iii) and induction, give a proof of the Cayley-Hamilton Theorem.
[Simple properties of determinants may be assumed without proof.]
-
Paper 4, Section II, G
2014 commentLet be a real vector space. What is the dual of If is a basis for , define the dual basis for , and show that it is indeed a basis for .
[No result about dimensions of dual spaces may be assumed.]
For a subspace of , what is the annihilator of ? If is -dimensional, how does the dimension of the annihilator of relate to the dimension of ?
Let be a linear map between finite-dimensional real vector spaces. What is the dual map ? Explain why the rank of is equal to the rank of . Prove that the kernel of is the annihilator of the image of , and also that the image of is the annihilator of the kernel of .
[Results about the matrices representing a map and its dual may be used without proof, provided they are stated clearly.]
Now let be the vector space of all real polynomials, and define elements of by setting to be the coefficient of in (for each ). Do the form a basis for ?
-
Paper 3, Section II, G
2014 commentLet be a nonsingular quadratic form on a finite-dimensional real vector space . Prove that we may write , where the restriction of to is positive definite, the restriction of to is negative definite, and for all and . [No result on diagonalisability may be assumed.]
Show that the dimensions of and are independent of the choice of and . Give an example to show that and are not themselves uniquely defined.
Find such a decomposition when and is the quadratic form
-
Paper 2, Section II, G
2014 commentDefine the determinant of an complex matrix . Explain, with justification, how the determinant of changes when we perform row and column operations on .
Let be complex matrices. Prove the following statements. (i) . (ii) .
-
Paper 4, Section I, H
2014 commentLet be a homogeneous Markov chain with state space and transition .
(a) Let Show that is a Markov chain and give its transition matrix. If , find in terms of the and the .
[Results from the course may be quoted without proof, provided they are clearly stated.]
(b) Suppose that and . Let , In terms of the , find
(i) and
(ii) .
What can you conclude about whether or not is a Markov chain?
-
Paper 3, Section I, H
2014 commentLet be a homogeneous Markov chain with state space . For in let denote the -step transition probability .
(i) Express the -step transition probability in terms of the -step and -step transition probabilities.
(ii) Write if there exists such that , and if and . Prove that if and then either both and are recurrent or both and are transient. [You may assume that a state is recurrent if and only if , and otherwise is transient.]
(iii) A Markov chain has state space and transition matrix
For each state , determine whether is recurrent or transient. [Results from the course may be quoted without proof, provided they are clearly stated.]
-
Paper 1, Section II, 20H
2014 commentConsider a homogeneous Markov chain with state space and transition . For a state , define the terms aperiodic, positive recurrent and ergodic.
Let and suppose that for we have and
where . Show that this Markov chain is irreducible.
Let be the first passage time to 0 . Find and show that state 0 is ergodic.
Find the invariant distribution for this Markov chain. Write down:
(i) the mean recurrence time for state ;
(ii) .
[Results from the course may be quoted without proof, provided they are clearly stated.]
-
Paper 2, Section II, H
2014 commentLet be a homogeneous Markov chain with state space and transition matrix . For , let
Prove that is the minimal non-negative solution to the equations
Three people and play a series of two-player games. In the first game, two people play and the third person sits out. Any subsequent game is played between the winner of the previous game and the person sitting out the previous game. The overall winner of the series is the first person to win two consecutive games. The players are evenly matched so that in any game each of the two players has probability of winning the game, independently of all other games. For , let be the ordered pair consisting of the winners of games and . Thus the state space is , and, for example, if wins the first game and wins the second.
The first game is between and . Treating and as absorbing states, or otherwise, find the probability of winning the series for each of the three players.
-
Paper 4, Section I, D
2014 commentConsider the ordinary differential equation
where is a positive constant and denotes the Dirac delta function. Physically relevant solutions for are bounded over the entire range .
(i) Find piecewise bounded solutions to this differential equations in the ranges and , respectively. [Hint: The equation for a constant may be solved using the Ansatz .]
(ii) Derive a matching condition at by integrating ( ) over the interval with and use this condition together with the requirement that be continuous at to determine the solution over the entire range .
-
Paper 2, Section I, D
2014 comment(i) Calculate the Fourier series for the periodic extension on of the function
defined on the interval .
(ii) Explain why the Fourier series for the periodic extension of can be obtained by term-by-term differentiation of the series for .
(iii) Let be the Fourier series for the periodic extension of . Determine the value of and explain briefly how it is related to the values of .
-
Paper 3, Section I, D
2014 commentUsing the method of characteristics, solve the differential equation
where and on .
-
Paper 1, Section II, D
2014 comment(a) Legendre's differential equation may be written
Show that for non-negative integer , this equation has a solution that is a polynomial of degree . Find and explicitly.
(b) Laplace's equation in spherical coordinates for an axisymmetric function (i.e. no dependence) is given by
Use separation of variables to find the general solution for .
Find the solution that satisfies the boundary conditions
where and are constants.
-
Paper 3, Section II, 15D
2014 commentLet be a linear second-order differential operator on the interval . Consider the problem
with bounded in .
(i) How is a Green's function for this problem defined?
(ii) How is a solution for this problem constructed from the Green's function?
(iii) Describe the continuity and jump conditions used in the construction of the Green's function.
(iv) Use the continuity and jump conditions to construct the Green's function for the differential equation
on the interval with the boundary conditions and an arbitrary bounded function . Use the Green's function to construct a solution for the particular case .
-
Paper 2, Section II, 16D
2014 commentThe Fourier transform of a function is defined as
A Green's function for the diffusion equation in one spatial dimension satisfies
(a) By applying a Fourier transform, show that the Fourier transform of this Green's function and the Green's function are
where is the Heaviside function. [Hint: The Fourier transform of a Gaussian , is given by
(b) The analogous result for the Green's function for the diffusion equation in two spatial dimensions is
Use this Green's function to construct a solution for to the diffusion equation
with the initial condition .
Now set
Find the solution for in terms of the exponential integral defined by
Use the approximation , valid for , to simplify this solution . Here is Euler's constant.
-
Paper 4, Section II, D
2014 commentLet be a complex-valued function defined on the interval and periodically extended to .
(i) Express as a complex Fourier series with coefficients . How are the coefficients obtained from ?
(ii) State Parseval's theorem for complex Fourier series.
(iii) Consider the function on the interval and periodically extended to for a complex but non-integer constant . Calculate the complex Fourier series of .
(iv) Prove the formula
(v) Now consider the case where is a real, non-integer constant. Use Parseval's theorem to obtain a formula for
What value do you obtain for this series for
-
Paper 3, Section I, E
2014 commentSuppose is a metric space. Do the following necessarily define a metric on ? Give proofs or counterexamples.
(i) for some constant , for all .
(ii) for all .
(iii) for all .
-
Paper 2, Section , E
2014 commentLet and be topological spaces. What does it mean to say that a function is continuous?
Are the following statements true or false? Give proofs or counterexamples.
(i) Every continuous function is an open map, i.e. if is open in then is open in .
(ii) If is continuous and bijective then is a homeomorphism.
(iii) If is continuous, open and bijective then is a homeomorphism.
-
Paper 1, Section II, E
2014 commentDefine what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.
Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if and are compact subspaces of a Hausdorff space then is compact.
A subset of is open in the cocountable topology if is empty or its complement in is countable. Is Hausdorff in the cocountable topology? Which subsets of are compact in the cocountable topology?
-
Paper 4, Section II, E
2014 commentExplain what it means for a metric space to be complete.
Let be a metric space. We say the subsets of , with , form a descending sequence in if .
Prove that the metric space is complete if and only if any descending sequence of non-empty closed subsets of , such that the diameters of the subsets converge to zero, has an intersection that is non-empty.
[Recall that the diameter of a set is the supremum of the set :
Give examples of
(i) a metric space , and a descending sequence of non-empty closed subsets of , with converging to 0 but .
(ii) a descending sequence of non-empty sets in with converging to 0 but .
(iii) a descending sequence of non-empty closed sets in with .
-
Paper 1, Section I,
2014 comment(i) A general multistep method for the numerical approximation to the scalar differential equation is given by
where . Show that this method is of order if and only if
where
(ii) A particular three-step implicit method is given by
where the are chosen to make the method third order. [The need not be found.] For what values of is the method convergent?
-
Paper 4, Section I, C
2014 commentConsider the quadrature given by
for , disjoint and . Show that it is not possible to make this quadrature exact for all polynomials of order .
For the case that and , by considering orthogonal polynomials find suitable and that make the quadrature exact on cubic polynomials.
[Hint: and ]
-
Paper 1, Section II, C
2014 commentDefine a Householder transformation and show that it is an orthogonal matrix. Briefly explain how these transformations can be used for QR factorisation of an matrix.
Using Householder transformations, find a QR factorisation of
Using this factorisation, find the value of for which
has a unique solution .
-
Paper 3, Section II, C
2014 commentA Runge-Kutta scheme is given by
for the solution of an autonomous differential equation , where is a real parameter. What is the order of the scheme? Identify all values of for which the scheme is A-stable. Determine the linear stability domain for this range.
-
Paper 2, Section II, C
2014 commentA linear functional acting on is approximated using a linear scheme . The approximation is exact when is a polynomial of degree . The error is given by . Starting from the Taylor formula for with an integral remainder term, show that the error can be written in the form
subject to a condition on that you should specify. Give an expression for .
Find and such that the approximation scheme
is exact for all that are polynomials of degree 2 . Assuming , apply the Peano kernel theorem to the error. Find and sketch for .
Find the minimum values for the constants and for which
and show explicitly that both error bounds hold for .
-
Paper 1, Section I, 8H
2014 commentState and prove the Lagrangian sufficiency theorem.
Use the Lagrangian sufficiency theorem to find the minimum of subject to (where and are real).
-
Paper 2, Section I, H
2014 commentExplain what is meant by a two-player zero-sum game with pay-off matrix , and state the optimal strategies for each player.
Find these optimal strategies when
-
Paper 4, Section II, H
2014 commentConsider a network with a single source and a single sink, where all the edge capacities are finite. Write down the maximum flow problem, and state the max-flow min-cut theorem.
Describe the Ford-Fulkerson algorithm. If all edge capacities are integers, explain why, starting from a suitable initial flow, the algorithm is guaranteed to end after a finite number of iterations.
The graph in the diagram below represents a one-way road network taking traffic from point to point via five roundabouts . The capacity of each road is shown on the diagram in terms of vehicles per minute. Assuming that all roundabouts can deal with arbitrary amounts of flow of traffic, find the maximum flow of traffic (in vehicles per minute) through this network of roads. Show that this flow is indeed optimal.
After a heavy storm, roundabout is flooded and only able to deal with at most 20 vehicles per minute. Find a suitable new network for the situation after the storm. Apply the Ford-Fulkerson algorithm to the new network, starting with the zero flow and explaining each step, to determine the maximum flow and the associated flows on each road.

-
Paper 3, Section II, H
2014 commentUse the two-phase simplex method to maximise subject to the constraints
Derive the dual of this linear programming problem and find the optimal solution of the dual.
-
Paper 4, Section I, A
2014 commentFor some quantum mechanical observable , prove that its uncertainty satisfies
A quantum mechanical harmonic oscillator has Hamiltonian
where . Show that (in a stationary state of energy )
Write down the Heisenberg uncertainty relation. Then, use it to show that
for our stationary state.
-
Paper 3, Section I, A
2014 commentThe wavefunction of a normalised Gaussian wavepacket for a particle of mass in one dimension with potential is given by
where . Given that is a solution of the time-dependent Schrödinger equation, find the complex-valued function and the real constant .
[You may assume that ]
-
Paper 1, Section II, A
2014 commentConsider a particle confined in a one-dimensional infinite potential well: for and for . The normalised stationary states are
where .
(i) Determine the and the stationary states' energies .
(ii) A state is prepared within this potential well: for , but for or . Find an explicit expansion of in terms of
(iii) If the energy of the state is then immediately measured, show that the probability that it is greater than is
where the are integers which you should find.
(iv) By considering the normalisation condition for in terms of the expansion in , show that
where and are integers which you should find.
-
Paper 3, Section II, A
2014 commentThe Hamiltonian of a two-dimensional isotropic harmonic oscillator is given by
where and denote position operators and and the corresponding momentum operators.
State without proof the commutation relations between the operators . From these commutation relations, write and in terms of a single operator. Now consider the observable
Ehrenfest's theorem states that, for some observable with expectation value ,
Use it to show that the expectation value of is constant with time.
Given two states
where and are constants, find a normalised linear combination of and that is an eigenstate of , and the corresponding eigenvalue. [You may assume that correctly normalises both and .] If a quantum state is prepared in the linear combination you have found at time , what is the expectation value of at a later time
-
Paper 2, Section II, A
2014 commentFor an electron of mass in a hydrogen atom, the time-independent Schrödinger equation may be written as
Consider normalised energy eigenstates of the form
where are orbital angular momentum eigenstates:
where and . The functions are normalised with
(i) Write down the resulting equation satisfied by , for fixed . Show that it has solutions of the form
where is a constant which you should determine. Show that
where is an integer which you should find (in terms of ). Also, show that
where and are integers that you should find in terms of .
(ii) Given the radius of the proton , show that the probability of the electron being found within the proton is approximately
finding the integer in terms of .
[You may assume that .]
-
Paper 1, Section I,
2014 commentConsider an estimator of an unknown parameter , and assume that for all . Define the bias and mean squared error of .
Show that the mean squared error of is the sum of its variance and the square of its bias.
Suppose that are independent identically distributed random variables with mean and variance , and consider estimators of of the form where .
(i) Find the value of that gives an unbiased estimator, and show that the mean squared error of this unbiased estimator is .
(ii) Find the range of values of for which the mean squared error of is smaller .
-
Paper 2, Section I, H
2014 commentThere are 100 patients taking part in a trial of a new surgical procedure for a particular medical condition. Of these, 50 patients are randomly selected to receive the new procedure and the remaining 50 receive the old procedure. Six months later, a doctor assesses whether or not each patient has fully recovered. The results are shown below:
\begin{tabular}{l|c|c} & Fully recovered & Not fully recovered \ \hline Old procedure & 25 & 25 \ \hline New procedure & 31 & 19 \end{tabular}
The doctor is interested in whether there is a difference in full recovery rates for patients receiving the two procedures. Carry out an appropriate significance level test, stating your hypotheses carefully. [You do not need to derive the test.] What conclusion should be reported to the doctor?
[Hint: Let denote the upper percentage point of a distribution. Then
-
Paper 4, Section II, H
2014 commentConsider a linear model
where is a known matrix, is a vector of unknown parameters and is an vector of independent random variables with unknown. Assume that has full rank . Find the least squares estimator of and derive its distribution. Define the residual sum of squares and write down an unbiased estimator of .
Suppose that and , for , where and are known with , and are independent random variables. Assume that at least two of the are distinct and at least two of the are distinct. Show that (where denotes transpose) may be written as in ( ) and identify and . Find in terms of the , and . Find the distribution of and derive a confidence interval for .
[Hint: You may assume that has a distribution, and that and the residual sum of squares are independent. Properties of distributions may be used without proof.]
-
Paper 1, Section II, H
2014 commentSuppose that , and are independent identically distributed Poisson random variables with expectation , so that
and consider testing against , where is a known value greater than 1. Show that the test with critical region is a likelihood ratio test of against . What is the size of this test? Write down an expression for its power.
A scientist counts the number of bird territories in randomly selected sections of a large park. Let be the number of bird territories in the th section, and suppose that are independent Poisson random variables with expectations respectively. Let be the area of the th section. Suppose that , and . Derive the generalised likelihood ratio for testing
What should the scientist conclude about the number of bird territories if is
[Hint: Let be where has a Poisson distribution with expectation . Then
-
Paper 3, Section II, H
2014 commentSuppose that are independent identically distributed random variables with
where is known and is an unknown parameter. Find the maximum likelihood estimator of .
Statistician 1 has prior density for given by , where . Find the posterior distribution for after observing data . Write down the posterior mean , and show that
where depends only on the prior distribution and is a constant in that is to be specified.
Statistician 2 has prior density for given by . Briefly describe the prior beliefs that the two statisticians hold about . Find the posterior mean and show that .
Suppose that increases (but and the remain unchanged). How do the prior beliefs of the two statisticians change? How does vary? Explain briefly what happens to and .
[Hint: The Beta distribution has density
with expectation and variance . Here, , is the Gamma function.]
-
Paper 1, Section I, C
2014 commentDefine the Legendre transform of a function where .
Show that for ,
Show that for where is a real, symmetric, invertible matrix with positive eigenvalues,
-
Paper 3, Section I,
2014 commentLet . Using Lagrange multipliers, find the location(s) and value of the maximum of on the intersection of the unit sphere and the ellipsoid given by .
-
Paper 4, Section II, H
2014 commentLet be a smooth projective curve of genus over an algebraically closed field of characteristic , and suppose there is a degree 2 morphism . How many ramification points of are there?
Suppose and are distinct ramification points of . Show that , but .
Now suppose . Show that every divisor of degree 2 on is linearly equivalent to for some , and deduce that every divisor of degree 0 is linearly equivalent to for some .
Show that the subgroup of the divisor class group of has order
-
Paper 3, Section II, H
2014 commentLet be a polynomial with distinct roots, , char , and let be the projective closure of the affine curve
Show that is smooth, with a single point at .
Pick an appropriate and compute the valuation for all .
Hence determine .
-
Paper 2, Section II, H
2014 comment(i) Let be an algebraically closed field, , and a subset of .
Let when . Show that is an ideal, and that does not have any non-zero nilpotent elements.
Let be affine varieties, and be a -algebra homomorphism. Show that determines a map of sets from to .
(ii) Let be an irreducible affine variety. Define the dimension of (in terms of the tangent spaces of ) and the transcendence dimension of .
State the Noether normalization theorem. Using this, or otherwise, prove that the transcendence dimension of equals the dimension of .
-
Paper 1, Section II, H
2014 commentLet be an algebraically closed field and . We say that is singular at if either is a singularity of the hypersurface or has an irreducible factor of multiplicity strictly greater than one with . Given , let and let
(i) Show that for some (you need not determine ) and that is a Zariski closed subvariety of .
(ii) Show that the fibres of the projection map are linear subspaces of . Conclude that .
(iii) Hence show that smooth is dense in .
[You may use standard results from lectures if they are accurately quoted.]
-
Paper 3, Section II, F
2014 commentLet be a simplicial complex in , which we may also consider as lying in using the first coordinates. Write . Show that if is a simplex of then is a simplex in .
Let be a subcomplex and let be the collection
of simplices in . Show that is a simplicial complex.
If is a Möbius band, and is its boundary, show that
-
Paper 4, Section II, F
2014 commentState the Lefschetz fixed point theorem.
Let be an orientable surface of genus (which you may suppose has a triangulation), and let be a continuous map such that
-
,
-
has no fixed points.
By considering the eigenvalues of the linear map , and their multiplicities, show that must be congruent to 1 modulo 3 .
-
-
Paper 2, Section II, F
2014 commentLet be a matrix with integer entries. Considering as the quotient space , show that the function
is well-defined and continuous. If in addition , show that is a homeomorphism.
State the Seifert-van Kampen theorem. Let be the space obtained by gluing together two copies of along their boundaries using the homeomorphism . Show that the fundamental group of is cyclic and determine its order.
-
Paper 1, Section II, F
2014 commentDefine what it means for a map to be a covering space. State the homotopy lifting lemma.
Let be a based covering space and let be a based map from a path-connected and locally path-connected space. Show that there is a based lift of if and only if .
-
Paper 4, Section II, A
2014 commentLet be a Bravais lattice in three dimensions. Define the reciprocal lattice .
State and prove Bloch's theorem for a particle moving in a potential obeying
Explain what is meant by a Brillouin zone for this potential and how it is related to the reciprocal lattice.
A simple cubic lattice is given by the set of points
where and are unit vectors parallel to the Cartesian coordinate axes in . A bodycentred cubic ) lattice is obtained by adding to the points at the centre of each cube, i.e. all points of the form
Show that is Bravais with primitive vectors
Find the reciprocal lattice . Hence find a consistent choice for the first Brillouin zone of a potential obeying
-
Paper 2, Section II, A
2014 comment(a) A classical particle of mass scatters on a central potential with energy , impact parameter , and scattering angle . Define the corresponding differential cross-section.
For particle trajectories in the Coulomb potential,
the impact parameter is given by
Find the differential cross-section as a function of and .
(b) A quantum particle of mass and energy scatters in a localised potential . With reference to the asymptotic form of the wavefunction at large , define the scattering amplitude as a function of the incident and outgoing wavevectors and (where ). Define the differential cross-section for this process and express it in terms of .
Now consider a potential of the form , where is a dimensionless coupling and does not depend on . You may assume that the Schrödinger equation for the wavefunction of a scattering state with incident wavevector may be written as the integral equation
where
Show that the corresponding scattering amplitude is given by
By expanding the wavefunction in powers of and keeping only the leading term, calculate the leading-order contribution to the differential cross-section, and evaluate it for the case of the Yukawa potential
By taking a suitable limit, obtain the differential cross-section for quantum scattering in the Coulomb potential defined in Part (a) above, correct to leading order in an expansion in powers of the constant . Express your answer as a function of the particle energy and scattering angle , and compare it to the corresponding classical cross-section calculated in Part (a).
Part II, List of Questions
[TURN OVER
-
Paper 3, Section II, A
2014 commentIn the nearly-free electron model a particle of mass moves in one dimension in a periodic potential of the form , where is a dimensionless coupling and has a Fourier series
with coefficients obeying for all .
Ignoring any degeneracies in the spectrum, the exact energy of a Bloch state with wavenumber can be expanded in powers of as
where is a normalised eigenstate of the free Hamiltonian with momentum and energy .
Working on a finite interval of length , where is a positive integer, we impose periodic boundary conditions on the wavefunction:
What are the allowed values of the wavenumbers and which appear in (1)? For these values evaluate the matrix element .
For what values of and does (1) cease to be a good approximation? Explain your answer. Quoting any results you need from degenerate perturbation theory, calculate to the location and width of the gaps between allowed energy bands for the periodic potential , in terms of the Fourier coefficients .
Hence work out the allowed energy bands for the following potentials:
-
Paper 1, Section II, A
2014 commentA particle of mass scatters on a localised potential well in one dimension. With reference to the asymptotic behaviour of the wavefunction as , define the reflection and transmission amplitudes, and , for a right-moving incident particle of wave number . Define also the corresponding amplitudes, and , for a left-moving incident particle of wave number . Derive expressions for and in terms of and .
(a) Define the -matrix, giving its elements in terms of and . Using the relation
(which you need not derive), show that the S-matrix is unitary. How does the S-matrix simplify if the potential well satisfies ?
(b) Consider the potential well
The corresponding Schrödinger equation has an exact solution
with energy , for every real value of . [You do not need to verify this.] Find the S-matrix for scattering on this potential. What special feature does the scattering have in this case?
(c) Explain the connection between singularities of the S-matrix and bound states of the potential well. By analytic continuation of the solution to appropriate complex values of , find the wavefunctions and energies of the bound states of the well. [You do not need to normalise the wavefunctions.]
-
Paper 3, Section II, J
2014 comment(i) Define a Poisson process with intensity . Specify without justification the distribution of . Let denote the jump times of . Derive the joint distribution of given .
(ii) Let be a Poisson process with intensity and let be a sequence of i.i.d. random variables, independent of , all having the same distribution as a random variable . Show that if is a real-valued function of real variables , and are the jump times of then
for all . [Hint: Condition on and , using (i).]
(iii) A university library is open from 9 am to . Students arrive at times of a Poisson process with intensity . Each student spends a random amount of time in the library, independently of the other students. These times are identically distributed for all students and have the same distribution as a random variable . Show that the number of students in the library at is a Poisson random variable with a mean that you should specify.
-
Paper 4, Section II, J
2014 comment(i) Define the queue with arrival rate and service rate . Find conditions on the parameters and for the queue to be transient, null recurrent, and positive recurrent, briefly justifying your answers. In the last case give with justification the invariant distribution explicitly. Answer the same questions for an queue.
(ii) At a taxi station, customers arrive at a rate of 3 per minute, and taxis at a rate of 2 per minute. Suppose that a taxi will wait no matter how many other taxis are present. However, if a person arriving does not find a taxi waiting he or she leaves to find alternative transportation.
Find the long-run proportion of arriving customers who get taxis, and find the average number of taxis waiting in the long run.
An agent helps to assign customers to taxis, and so long as there are taxis waiting he is unable to have his coffee. Once a taxi arrives, how long will it take on average before he can have another sip of his coffee?
-
Paper 1, Section II, J
2014 comment(i) Explain what a -matrix is. Let be a -matrix. Define the notion of a Markov chain in continuous time with -matrix given by , and give a construction of . [You are not required to justify this construction.]
(ii) A population consists of individuals at time . We assume that each individual gives birth to a new individual at constant rate . As the population is competing for resources, we assume that for each , if , then any individual in the population at time dies in the time interval with probability , where is a given sequence satisfying for . Formulate a Markov chain model for and write down the -matrix explicitly. Then find a necessary and sufficient condition on so that the Markov chain has an invariant distribution. Compute the invariant distribution in the case where and .
-
Paper 2, Section II, J
2014 comment(i) Explain what the Moran model and the infinite alleles model are. State Ewens' sampling formula for the distribution of the allelic frequency spectrum in terms of where with denoting the mutation rate per individual and the population size.
Let be the number of allelic types in a sample of size . Give, without justification, an expression for in terms of .
(ii) Let and be as above. Show that for we have that
for some constant that does not depend on .
Show that, given , the distribution of the allelic frequency spectrum does not depend on .
Show that the value of which maximises is the one for which .
-
Paper 4, Section II, C
2014 commentDerive the leading-order Liouville Green (or WKBJ) solution for to the ordinary differential equation
where .
The function satisfies the ordinary differential equation
subject to the boundary condition . Show that the Liouville-Green solution of (1) for takes the asymptotic forms
where and are constants.
Hint: You may assume that
Explain, showing the relevant change of variables, why the leading-order asymptotic behaviour for can be obtained from the reduced equation
The unique solution to with is , where the Bessel function is known to have the asymptotic form
Hence find the values of and .
-
Paper 3, Section II, C
2014 comment(a) Find the Stokes ray for the function as with , where
(b) Describe how the leading-order asymptotic behaviour as of
may be found by the method of stationary phase, where and are real functions and the integral is taken along the real line. You should consider the cases for which:
(i) is non-zero in and has a simple zero at .
(ii) is non-zero apart from having one simple zero at , where .
(iii) has more than one simple zero in with and .
Use the method of stationary phase to find the leading-order asymptotic form as of
[You may assume that
-
Paper 1, Section II, C
2014 comment(a) Consider the integral
Suppose that possesses an asymptotic expansion for of the form
where are constants. Derive an asymptotic expansion for as in the form
giving expressions for and in terms of and the gamma function. Hence establish the asymptotic approximation as
where .
(b) Using Laplace's method, or otherwise, find the leading-order asymptotic approximation as for
[You may assume that for ,
-
Paper 4, Section I, A
2014 commentConsider a heavy symmetric top of mass with principal moments of inertia , and , where . The top is pinned at point , which is at a distance from the centre of mass, , as shown in the figure.

Its angular velocity in a body frame is given by
where and are the Euler angles.
(a) Assuming that , are chosen to be the principal axes, write down the Lagrangian of the top in terms of and the principal moments of inertia. Hence find the Lagrangian in terms of the Euler angles.
(b) Find all conserved quantities. Show that , the spin of the top, is constant.
(c) By eliminating and , derive a second-order differential equation for .
-
Paper 3, Section I, A
2014 comment(a) The action for a one-dimensional dynamical system with a generalized coordinate and Lagrangian is given by
State the principle of least action. Write the expression for the Hamiltonian in terms of the generalized velocity , the generalized momentum and the Lagrangian . Use it to derive Hamilton's equations from the principle of least action.
(b) The motion of a particle of charge and mass in an electromagnetic field with scalar potential and vector potential is characterized by the Lagrangian
(i) Write down the Hamiltonian of the particle.
(ii) Consider a particle which moves in three dimensions in a magnetic field with , where is a constant. There is no electric field. Obtain Hamilton's equations for the particle.
-
Paper 2, Section I, A
2014 commentThe components of the angular velocity of a rigid body and of the position vector are given in a body frame.
(a) The kinetic energy of the rigid body is defined as
Given that the centre of mass is at rest, show that can be written in the form
where the explicit form of the tensor should be determined.
(b) Explain what is meant by the principal moments of inertia.
(c) Consider a rigid body with principal moments of inertia and , which are all unequal. Derive Euler's equations of torque-free motion
(d) The body rotates about the principal axis with moment of inertia . Derive the condition for stable rotation.
-
Paper 1, Section I, A
2014 commentConsider a one-dimensional dynamical system with generalized coordinate and momentum .
(a) Define the Poisson bracket of two functions and .
(b) Verify the Leibniz rule
(c) Explain what is meant by a canonical transformation .
(d) State the condition for a transformation to be canonical in terms of the Poisson bracket . Use this to determine whether or not the following transformations are canonical:
(i) ,
(ii) ,
(iii) .
-
Paper 4, Section II, A
2014 comment(a) Consider a system with one degree of freedom, which undergoes periodic motion in the potential . The system's Hamiltonian is
(i) Explain what is meant by the angle and action variables, and , of the system and write down the integral expression for the action variable . Is conserved? Is conserved?
(ii) Consider , where is a positive constant. Find in terms of , the total energy , the mass , and a dimensionless constant factor (which you need not compute explicitly).
(iii) Hence describe how changes with if varies slowly with time. Justify your answer.
(b) Consider now a particle which moves in a plane subject to a central force-field .
(i) Working in plane polar coordinates , write down the Hamiltonian of the system. Hence deduce two conserved quantities. Prove that the system is integrable and state the number of action variables.
(ii) For a particle which moves on an elliptic orbit find the action variables associated with radial and tangential motions. Can the relationship between the frequencies of the two motions be deduced from this result? Justify your answer.
(iii) Describe how changes with and if one or both of them vary slowly with time.
[You may use
where .]
-
Paper 2, Section II, A
2014 commentA planar pendulum consists of a mass at the end of a light rod of length . The pivot of the pendulum is attached to a bead of mass , which slides along a horizontal rod without friction. The bead is connected to the ends of the horizontal rod by two identical springs of force constant . The pivot constrains the pendulum to swing in the vertical plane through the horizontal rod. The horizontal rod is mounted on a bracket, so the system could rotate about the vertical axis which goes through its centre as shown in the figure.

(a) Initially, the system is not allowed to rotate about the vertical axis.
(i) Identify suitable generalized coordinates and write down the Lagrangian of the system.
(ii) Write down expression(s) for any conserved quantities. Justify your answer.
(iii) Derive the equations of motion.
(iv) For and , find the frequencies of small oscillations around the stable equilibrium and the corresponding normal modes. Describe the respective motions of the system.
(b) Assume now that the system is free to rotate about the vertical axis without friction. Write down the Lagrangian of the system. Identify and calculate the additional conserved quantity.
-
Paper 4, Section I, I
2014 commentExplain what is meant by a Bose-Ray Chaudhuri-Hocquenghem (BCH) code with design distance . Prove that, for such a code, the minimum distance between code words is at least . How many errors will the code detect? How many errors will it correct?
-
Paper 3, Section I, I
2014 commentLet be a random variable that takes values in the finite alphabet . Prove that there is a decodable binary code that satisfies
where is the length of the code word and is the entropy of .
Is it always possible to find such a code with Justify your answer.
-
Paper 2, Section , I
2014 commentLet be a decodable binary code defined on a finite alphabet . Let be the length of the code word . Prove that
Show that, for the decodable code described above, there is a prefixfree code with each code word having length . [You may use, without proof, any standard results from the course.]
-
Paper 1, Section I,
2014 commentState and prove Gibbs' inequality.
Show that, for a pair of discrete random variables and , each taking finitely many values, the joint entropy satisfies
with equality precisely when and are independent.
-
Paper 2, Section II, I
2014 commentWhat is the information capacity of a memoryless, time-independent channel? Compute the information capacity of a binary symmetric channel with probability of error. Show the steps in your computation.
Binary digits are transmitted through a noisy channel, which is memoryless and time-independent. With probability the digit is corrupted and noise is received, otherwise the digit is transmitted unchanged. So, if we denote the input by 0 and 1 and the output as and 1 with denoting the noise, the transition matrix is
Compute the information capacity of this channel.
Explain how to code a message for transmission through the channel described above, and how to decode it, so that the probability of error for each bit is arbitrarily small.
-
Paper 1, Section II, I
2014 commentDescribe, briefly, either the RSA or the Elgamal public key cipher. You should explain, without proof, why it is believed to be difficult to break the cipher you describe.
How can such a cipher be used to sign messages? You should explain how the intended recipient of the message can (a) know from whom it came; (b) know that the message has not been changed; and (c) demonstrate that the sender must have signed it.
Let be friendly individuals each of whom has a public key cipher. wishes to send a message to by passing it first to , then passes it to to , until finally it is received by . At each stage the message can be modified to show from whom it was received and to whom it is sent. Devise a way in which these modifications can be made so that can be confident both of the content of the original message and that the message has been passed through the intermediaries in that order and has not been modified by an enemy agent. Assume that it takes a negligible time to transmit a message from to for each , but the time needed to modify a message is not negligible.
-
Paper 4, Section I, E
2014 commentA homogeneous and isotropic universe, with cosmological constant , has expansion scale factor and Hubble expansion rate . The universe contains matter with density and pressure which satisfy the positive-energy condition . The acceleration equation is
If , show that
Deduce that and at a finite time in the past or the future. What property of distinguishes the two cases?
Give a simple counterexample with to show that this deduction fails to hold when .
-
Paper 3, Section I, E
2014 commentConsider a finite sphere of zero-pressure material of uniform density which expands with radius , where is an arbitary constant, due to the evolution of the expansion scale factor . The sphere has constant total mass and its radius satisfies
where
with constant. Show that the scale factor obeys the equation
where is a constant. Explain why the sign, but not the magnitude, of is important. Find exact solutions of this equation for when
(i) and ,
(ii) and ,
(iii) and .
Which two of the solutions (i)-(iii) are relevant for describing the evolution of the universe after the radiation-dominated era?
-
Paper 2, Section I, E
2014 commentA self-gravitating fluid with density , pressure and velocity in a gravitational potential obeys the equations
Assume that there exists a static constant solution of these equations with and , for which can be neglected. This solution is perturbed. Show that, to first order in the perturbed quantities, the density perturbations satisfy
where and . Show that there are solutions to this equation of the form
where and are constants and
Interpret these solutions physically in the limits of small and large , explaining what happens to density perturbations on large and small scales, and determine the critical wavenumber that divides the two distinct behaviours of the perturbation.
-
Paper 1, Section I, E
2014 commentWhich particle states are expected to be relativistic and which interacting when the temperature of the early universe satisfies (i) , (ii) , (iii) ?
Calculate the total spin weight factor, , of the relativistic particles and the total spin weight factor, , of the interacting particles, in each of the three temperature intervals.
What happens when the temperature falls below Calculate the ratio of the temperatures of neutrinos and photons. Find the effective value of after the universe cools below this temperature. [Note that the equilibrium entropy density is given by , where is the density and is the pressure.]
-
Paper 3, Section II, E
2014 commentThe luminosity distance to an astronomical light source is given by , where is the expansion scale factor and is the comoving distance in the universe defined by . A zero-curvature Friedmann universe containing pressure-free matter and a cosmological constant with density parameters and , respectively, obeys the Friedmann equation
where is the Hubble expansion rate of the universe and the subscript 0 denotes present-day values, with .
If is the redshift, show that
Find when and when . Roughly sketch the form of for these two cases. What is the effect of a cosmological constant on the luminosity distance at a fixed value of ? Briefly describe how the relation between luminosity distance and redshift has been used to establish the acceleration of the expansion of the universe.
-
Paper 1, Section II, E
2014 commentWhat are the cosmological flatness and horizon problems? Explain what form of time evolution of the cosmological expansion scale factor must occur during a period of inflationary expansion in a Friedmann universe. How can inflation solve the horizon and flatness problems? [You may assume an equation of state where pressure is proportional to density .]
The universe has Hubble expansion rate and contains only a scalar field with self-interaction potential . The density and pressure are given by
in units where . Show that the conservation equation
requires
If the Friedmann equation has the form
and the scalar-field potential has the form
where and are positive constants, show that there is an exact cosmological solution with
where is a constant. Find the algebraic relation between and . Show that a solution only exists when . For what range of values of does inflation occur? Comment on what happens when .
-
Paper 4, Section II, G
2014 commentLet be a closed interval, smooth real valued functions on with strictly positive at all points, and a positively oriented orthonormal triad of vectors in . An application of the fundamental theorem on the existence of solutions to ODEs implies that there exists a unique smooth family of triples of vectors for satisfying the differential equations
with initial conditions and , and that forms a positively oriented orthonormal triad for all . Assuming this fact, consider defined by ; show that defines a smooth immersed curve parametrized by arc-length, which has curvature and torsion given by and , and that is uniquely determined by this property up to rigid motions of . Prove that is a plane curve if and only if is identically zero.
If , calculate the curvature and torsion of the smooth curve given by
Suppose now that is a smooth simple closed curve parametrized by arc-length with curvature everywhere positive. If both and are constant, show that and . If is constant and is not identically zero, show that . Explain what it means for to be knotted; if is knotted and is constant, show that for some . [You may use standard results from the course if you state them precisely.]
-
Paper 3, Section II, G
2014 commentLet be a parametrized curve on a smooth embedded surface . Define what is meant by a vector field along and the concept of such a vector field being parallel. If and are both parallel vector fields along , show that the inner product is constant.
Given a local parametrization , define the Christoffel symbols on . Given a vector , prove that there exists a unique parallel vector field along with (recall that is called the parallel transport of along ).
Suppose now that the image of also lies on another smooth embedded surface and that for all . Show that parallel transport of a vector is the same whether calculated on or . Suppose is the unit sphere in with centre at the origin and let be the curve on given by
for some fixed angle . Suppose is the unit tangent vector to at and let be its image in under parallel transport along . Show that the angle between and is .
[Hint: You may find it useful to consider the circular cone which touches the sphere along the curve .]
-
Paper 2, Section II, G
2014 commentDefine the terms Gaussian curvature and mean curvature for a smooth embedded oriented surface . [You may assume the fact that the derivative of the Gauss map is self-adjoint.] If at all points of , show that both and are locally constant. [Hint: Use the symmetry of second partial derivatives of the field of unit normal vectors.]
If at all points of , show that the unit normal vector to is locally constant and that is locally contained in a plane. If is a strictly positive constant on and is a local parametrization (where is connected) on with unit normal vector for , show that is constant on . Deduce that is locally contained in a sphere of radius .
If is connected with at all points of , deduce that is contained in either a plane or a sphere.
-
Paper 1, Section II, G
2014 commentDefine the concepts of (smooth) manifold and manifold with boundary for subsets of .
Let be the subset defined by the equations
Prove that is a manifold of dimension four.
For , let denote the spherical ball . Prove that is empty if , is a manifold diffeomorphic to if , and is a manifold with boundary if , with each component of the boundary diffeomorphic to .
[You may quote without proof any general results from lectures that you may need.]
-
Paper 4, Section I, D
2014 commentConsider the map for . What is the maximum value, , for which the interval is mapped into itself?
Analyse the first two bifurcations that occur as increases from 0 towards , including an identification of the values of at which the bifurcation occurs and the type of bifurcation.
What type of bifurcation do you expect as the third bifurcation? Briefly give your reasoning.
-
Paper 3, Section , D
2014 commentDefine the Poincaré index of a closed curve for a vector field .
Explain carefully why the index of is fully determined by the fixed points of the dynamical system that lie within .
What is the Poincaré index for a closed curve if it (a) encloses only a saddle point, (b) encloses only a focus and (c) encloses only a node?
What is the Poincaré index for a closed curve that is a periodic trajectory of the dynamical system?
A dynamical system in has 2 saddle points, 1 focus and 1 node. What is the maximum number of different periodic orbits? [For the purposes of this question, two orbits are said to be different if they enclose different sets of fixed points.]
-
Paper 2, Section I, D
2014 commentConsider the system
Show that when the fixed point at the origin has a stationary bifurcation.
Find the centre subspace of the extended system linearised about .
Find an approximation to the centre manifold giving as a function of and , including terms up to quadratic order.
Hence deduce an expression for on the centre manifold, and identify the type of bifurcation at .
-
Paper 1, Section I, D
2014 commentConsider the system
Show that the origin is a hyperbolic fixed point and find the stable and unstable invariant subspaces of the linearised system.
Calculate the stable and unstable manifolds correct to quadratic order, expressing as a function of for each.
-
Paper 4, Section II, D
2014 commentA dynamical system has a fixed point at the origin. Define the terms Lyapunov stability, asymptotic stability and Lyapunov function with respect to this fixed point. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.
(a) Consider the system
Construct a Lyapunov function of the form . Deduce that the origin is asymptotically stable, explaining your reasoning carefully. Find the greatest value of such that use of this Lyapunov function guarantees that the trajectory through approaches the origin as .
(b) Consider the system
Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region .
-
Paper 3, Section II, D
2014 commentLet be a continuous one-dimensional map of an interval . Explain what is meant by saying that has a horseshoe.
A map on the interval is a tent map if
(i) and ;
(ii) for some with is linear and increasing on the interval , linear and decreasing on the interval , and continuous at .
Consider the tent map defined on the interval by
with . Find the corresponding expressions for .
Find the non-zero fixed point and the points that satisfy
Sketch graphs of and showing the points corresponding to and . Indicate the values of and at their maxima and minima and also the gradients of each piece of their graphs.
Identify a subinterval of on which is a tent map. Hence demonstrate that has a horseshoe if .
Explain briefly why has a horseshoe when .
Why are there periodic points of arbitrarily close to for , but no such points for ? Explain carefully any results or terms that you use.
-
Paper 4, Section II, 35C
2014 comment(i) The action for a point particle of rest mass and charge moving along a trajectory in the presence of an electromagnetic 4 -vector potential is
where is an arbitrary parametrization of the path and is the Minkowski metric. By varying the action with respect to , derive the equation of motion , where and overdots denote differentiation with respect to proper time for the particle.
(ii) The particle moves in constant electric and magnetic fields with non-zero Cartesian components and , with in some inertial frame. Verify that a suitable 4-vector potential has components
in that frame.
Find the equations of motion for and in terms of proper time . For the case of a particle that starts at rest at the spacetime origin at , show that
Find the trajectory and sketch its projection onto the plane.
-
Paper 3, Section II, C
2014 commentThe 4-vector potential (in the Lorenz gauge ) due to a localised source with conserved 4-vector current is
where . For a source that varies slowly in time, show that the spatial components of at a distance that is large compared to the spatial extent of the source are
where is the electric dipole moment of the source, which you should define. Explain what is meant by the far-field region, and calculate the leading-order part of the magnetic field there.
A point charge moves non-relativistically in a circle of radius in the plane with angular frequency (such that ). Show that the magnetic field in the far-field at the point with spherical polar coordinates and has components along the and directions given by
Calculate the total power radiated by the charge.
-
Paper 1, Section II, 36C
2014 comment(i) Starting from the field-strength tensor , where is the 4-vector potential with components such that
derive the transformation laws for the components of the electric field and the magnetic field under the standard Lorentz boost with
(ii) Two point charges, each with electric charge , are at rest and separated by a distance in some inertial frame . By transforming the fields from the rest frame , calculate the magnitude and direction of the force between the two charges in an inertial frame in which the charges are moving with speed in a direction perpendicular to their separation.
(iii) The 4-force for a particle with 4-momentum is , where is proper time. Show that the components of in an inertial frame in which the particle has 3 -velocity are
where and is the 3-force acting on the particle. Hence verify that your result in (ii) above is consistent with Lorentz transforming the electromagnetic 3 -force from the rest frame .
-
Paper 4, Section II, B
2014 commentAn incompressible fluid of density and kinematic viscosity is confined in a channel with rigid stationary walls at . A spatially uniform pressure gradient is applied in the -direction. What is the physical significance of the dimensionless number
Assuming that the flow is unidirectional and time-harmonic, obtain expressions for the velocity profile and the total flux. [You may leave your answers as the real parts of complex functions.]
In each of the limits and , find and sketch the flow profiles, find leading-order asymptotic expressions for the total flux, and give a physical interpretation.
Suppose now that and that the channel walls oscillate in their own plane with velocity in the -direction. Without explicit calculation of the solution, sketch the flow profile in each of the limits and .
-
Paper 2, Section II, B
2014 commentAir is blown over the surface of a large, deep reservoir of water in such a way as to exert a tangential stress in the -direction of magnitude for , with . The water is otherwise at rest and occupies the region . The surface remains flat.
Find order-of-magnitude estimates for the boundary-layer thickness and tangential surface velocity in terms of the relevant physical parameters.
Using the boundary-layer equations, find the ordinary differential equation governing the dimensionless function defined in the streamfunction
What are the boundary conditions on ?
Does as ? Why, or why not?
The total horizontal momentum flux across the vertical line is proportional to for . Find the exponent . By considering the steadiness of the momentum balance in the region , explain why the value of is consistent with the form of the stress exerted on the boundary.
-
Paper 3, Section II, B
2014 commentA rigid sphere of radius falls under gravity through an incompressible fluid of density and viscosity towards a rigid horizontal plane. The minimum gap between the sphere and the plane satisfies . Find an approximation for the gap thickness between the sphere and the plane in the region , where is the distance from the axis of symmetry.
For a prescribed value of , use lubrication theory to find the radial velocity and the fluid pressure in the region . Explain why the approximations of lubrication theory require and .
Calculate the total vertical force due to the motion that is exerted by the fluid on the sphere. Deduce that if the sphere is settling under its own weight (corrected for buoyancy) then decreases exponentially. What is the exponential decay rate for a solid sphere of density in a fluid of density ?
-
Paper 1, Section II, B
2014 commentA particle of arbitrary shape and volume moves at velocity through an unbounded incompressible fluid of density and viscosity . The Reynolds number of the flow is very small so that the inertia of the fluid can be neglected. Show that the particle experiences a force due to the surface stresses given by
where is a dimensionless second-rank tensor determined solely by the shape and orientation of the particle. State the reason why must be positive definite.
Show further that, if the particle has the same reflectional symmetries as a cube, then
Let be the radius of the smallest sphere that contains the particle (still assuming cubic symmetry). By considering the Stokes flow associated with this sphere, suitably extended, and using the minimum dissipation theorem (which should be stated carefully), show that
[You may assume the expression for the Stokes drag on a sphere.]
-
Paper 4, Section I, B
2014 commentLet be a function such that
where and is not real. Show that if is analytic on then it is a constant. [Liouville's theorem may be used if stated.] Give an example of a non-constant meromorphic function which satisfies (1).
-
Paper 3, Section I, B
2014 commentState the conditions for a point to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.
Find all singular points of the Airy equation
and determine whether they are regular or irregular.
-
Paper 1, Section , B
2014 commentShow that the Cauchy-Riemann equations for are equivalent to
where , and should be defined in terms of and . Use Green's theorem, together with the formula , to establish the generalised Cauchy formula
where is a contour in the complex plane enclosing the region and is sufficiently differentiable.
-
Paper 2, Section I, B
2014 commentSuppose is a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane. Define the monodromy matrix around .
Demonstrate that if
then the differential equation admits a solution of the form , where and are single-valued functions.
-
Paper 2, Section II, 14B
2014 commentUse the Euler product formula
to show that:
(i) ;
(ii) , where .
Deduce that
-
Paper 1, Section II, 14B
2014 commentObtain solutions of the second-order ordinary differential equation
in the form
where the function and the choice of contour should be determined from the differential equation.
Show that a non-trivial solution can be obtained by choosing to be a suitable closed contour, and find the resulting solution in this case, expressing your answer in the form of a power series.
Describe a contour that would provide a second linearly independent solution for the case .
-
Paper 4, Section II, H
2014 comment(i) Let be a finite subgroup of the multiplicative group of a field. Show that is cyclic.
(ii) Let be the th cyclotomic polynomial. Let be a prime not dividing , and let be a splitting field for over . Show that has elements, where is the least positive integer such that .
(iii) Find the degrees of the irreducible factors of over , and the number of factors of each degree.
-
Paper 3, Section II, H
2014 commentLet be an algebraic extension of fields, and . What does it mean to say that is separable over ? What does it mean to say that is separable?
Let be the field of rational functions over .
(i) Show that if is inseparable over then contains a th root of .
(ii) Show that if is finite there exists and such that and is separable.
Show that is an irreducible separable polynomial over the field of rational functions . Find the degree of the splitting field of over .
-
Paper 2, Section II, H
2014 commentDescribe the Galois correspondence for a finite Galois extension .
Let be the splitting field of over . Compute the Galois group of . For each subgroup of , determine the corresponding subfield of .
Let be a finite Galois extension whose Galois group is isomorphic to . Show that is the splitting field of a separable polynomial of degree .
-
Paper 1, Section II, 18H
2014 commentWhat is meant by the statement that is a splitting field for
Show that if , then there exists a splitting field for over . Explain the sense in which a splitting field for over is unique.
Determine the degree of a splitting field of the polynomial over in the cases (i) , (ii) , and (iii) .
-
Paper 4, Section II, E
2014 commentA plane-wave spacetime has line element
where . Show that the line element is unchanged by the coordinate transformation
Show more generally that the line element is unchanged by coordinate transformations of the form
where and are functions of , which you should determine and which depend in total on four parameters (arbitrary constants of integration).
Deduce (without further calculation) that the line element is unchanged by a 6parameter family of coordinate transformations, of which a 5 -parameter family leave invariant the surfaces constant.
For a general coordinate transformation , give an expression for the transformed Ricci tensor in terms of the Ricci tensor and the transformation matrices . Calculate when the transformation is given by and deduce that
-
Paper 2, Section II, E
2014 commentShow how the geodesic equations and hence the Christoffel symbols can be obtained from a Lagrangian.
In units with , the FLRW spacetime line element is
Show that .
You are given that, for the above metric, and , where is the Einstein tensor, which is diagonal. Verify by direct calculation that .
Solve the vacuum Einstein equations in the presence of a cosmological constant to determine the form of .
-
Paper 3, Section II,
2014 commentThe vector field is the normalised tangent to a congruence of timelike geodesics, and .
Show that:
(i) ;
(ii) .
[You may use the Ricci identity .]
Now assume that is symmetric and let . By writing , or otherwise, show that
where and . [You may use without proof the result that \left.\widetilde{B}_{a b} \widetilde{B}^{a b} \geqslant 0 .\right]
Assume, in addition, that the stress-energy tensor takes the perfect-fluid form and that . Show that
and deduce that, if , then will become unbounded for some value of less than .
-
Paper 1, Section II, 37E
2014 commentFor a timelike geodesic in the equatorial plane of the Schwarzschild spacetime with line element
derive the equation
where
and and are constants. The dot denotes the derivative with respect to an affine parameter satisfying .
Given that there is a stable circular orbit at , show that
where .
Compute , the orbital angular frequency (with respect to ).
Show that the angular frequency of small radial perturbations is given by
Deduce that the rate of precession of the perihelion of the Earth's orbit, , is approximately , where is the time taken for light to travel from the Sun to the Earth. [You should assume that the Earth's orbit is approximately circular, with and
-
Paper 4, Section I, F
2014 commentDefine the limit set of a Kleinian group . Assuming that has no finite orbit in , and that , prove that if is any non-empty closed set which is invariant under , then .
-
Paper 3, Section I, F
2014 commentLet denote the hyperbolic plane, and be a non-degenerate triangle, i.e. the bounded region enclosed by three finite-length geodesic arcs. Prove that the three angle bisectors of meet at a point.
Must the three vertices of lie on a hyperbolic circle? Justify your answer.
-
Paper 2, Section I, F
2014 commentLet be non-identity Möbius transformations. Prove that and commute if and only if one of the following holds:
-
;
-
are involutions each of which exchanges the other's fixed points.
Give an example to show that the second case can occur.
-
-
Paper 1, Section I, F
2014 commentLet be a finite group. Suppose does not preserve any plane in . Show that for any point in the unit sphere , the stabiliser contains at most 5 elements.
-
Paper 1, Section II, F
2014 commentProve that an orientation-preserving isometry of the ball-model of hyperbolic space which fixes the origin is an element of . Hence, or otherwise, prove that a finite subgroup of the group of orientation-preserving isometries of hyperbolic space has a common fixed point.
Can an infinite non-cyclic subgroup of the isometry group of have a common fixed point? Can any such group be a Kleinian group? Justify your answers.
-
Paper 4, Section II, F
2014 commentDefine the -dimensional Hausdorff measure of a set . Explain briefly how properties of this measure may be used to define the Hausdorff dimension of such a set.
Prove that the limit sets of conjugate Kleinian groups have equal Hausdorff dimension. Hence, or otherwise, prove that there is no subgroup of which is conjugate in to .
-
Paper 4, Section II, I
2014 commentDefine the Ramsey number . What is the value of ? Prove that holds for and deduce that exists.
Show that and that .
Show that . [Hint: For the lower bound, choose a suitable subset and colour e red if is odd.]
-
Paper 3, Section II, I
2014 commentProve that for every graph . Prove further that, if , then unless is complete.
Let . A graph is said to be -critical if , but for every vertex of . Show that, if is -critical, then .
Let , and let be the graph with an edge removed. Show that has the following property: it has two vertices which receive the same colour in every -colouring of . By considering two copies of , construct a -colourable graph of order with the following property: it has three vertices which receive the same colour in every -colouring of .
Construct, for all integers and , a -critical graph of order with
-
Paper 2, Section II, I
2014 commentLet and be integers with . Show that every connected graph of order , in which for every pair of non-adjacent vertices, contains a path of length .
Let and be integers with . Show that a graph of order that contains no path of length has at most edges, and that this value is achieved only if divides and is the union of disjoint copies of . [Hint: Proceed by induction on and consider a vertex of minimum degree.]
-
Paper 1, Section II, I
2014 commentShow that a graph is bipartite if and only if all of its cycles are of even length.
Show that a bridgeless plane graph is bipartite if and only if all of its faces are of even length.
Let be an Eulerian plane graph. Show that the faces of can be coloured with two colours so that no two contiguous faces have the same colour. Deduce that it is possible to assign a direction to each edge of in such a way that the edges around each face form a directed cycle.
-
Paper 3, Section II, D
2014 commentWhat does it mean to say that a finite-dimensional Hamiltonian system is integrable? State the Arnold-Liouville theorem.
A six-dimensional dynamical system with coordinates is governed by the differential equations
for , where are positive constants. Show that these equations can be written in the form
for an appropriate function . By introducing the coordinates
show that the system can be written in Hamiltonian form
for some Hamiltonian which you should determine.
Show that the three functions
are first integrals of the Hamiltonian system.
Making use of the fundamental Poisson brackets and , show that
Hence show that the Hamiltonian system is integrable.
-
Paper 2, Section II, D
2014 commentLet be a smooth function that decays rapidly as and let denote the associated Schrödinger operator. Explain very briefly each of the terms appearing in the scattering data
associated with the operator . What does it mean to say is reflectionless?
Given , define the function
If is the unique solution to the GLM equation
what is the relationship between and ?
Now suppose that is time dependent and that it solves the KdV equation . Show that obeys the Lax equation
Show that the discrete eigenvalues of are time independent.
In what follows you may assume the time-dependent scattering data take the form
Show that if is reflectionless, then the solution to the KdV equation takes the form
where is an matrix which you should determine.
Assume further that , where is smooth and decays rapidly at infinity. Show that, for any fixed ,
Comment briefly on the significance of this result.
[You may assume for a non-singular matrix .]
-
Paper 1, Section II, D
2014 commentConsider the coordinate transformation
Show that defines a one-parameter group of transformations. Define what is meant by the generator of a one-parameter group of transformations and compute it for the above case.
Now suppose . Explain what is meant by the first prolongation of . Compute in this case and deduce that
Similarly find .
Define what is meant by a Lie point symmetry of the first-order differential equation . Describe this condition in terms of the vector field that generates the Lie point symmetry. Consider the case
where is an arbitrary smooth function of one variable. Using , show that generates a Lie point symmetry of the corresponding differential equation.
-
Paper 3, Section II, G
2014 comment(i) State carefully the theorems of Stone-Weierstrass and Arzelá-Ascoli (work with real scalars only).
(ii) Let denote the family of functions on of the form
where the are real and for all . Prove that any sequence in has a subsequence that converges uniformly on .
(iii) Let be a continuous function such that and exists. Show that for each there exists a real polynomial having only odd powers, i.e. of the form
such that . Show that the same holds without the assumption that is differentiable at 0 .
-
Paper 1, Section II, G
2014 commentLet and be normed spaces. What is an isomorphism between and ? Show that a bounded linear map is an isomorphism if and only if is surjective and there is a constant such that for all . Show that if there is an isomorphism and is complete, then is complete.
Show that two normed spaces of the same finite dimension are isomorphic. [You may assume without proof that any two norms on a finite-dimensional space are equivalent.] Briefly explain why this implies that every finite-dimensional space is complete, and every closed and bounded subset of a finite-dimensional space is compact.
Let and be subspaces of a normed space with . Assume that is closed in and is finite-dimensional. Prove that is closed in . [Hint: First show that the function restricted to the unit sphere of F achieves its minimum.]
-
Paper 2, Section II, G
2014 comment(a) Let and be Banach spaces, and let be a surjective linear map. Assume that there is a constant such that for all . Show that is continuous. [You may use any standard result from general Banach space theory provided you clearly state it.] Give an example to show that the assumption that and are complete is necessary.
(b) Let be a closed subset of a Banach space such that
(i) for each ;
(ii) for each and ;
(iii) for each , there exist such that .
Prove that, for some , the unit ball of is contained in the closure of the set
[You may use without proof any version of the Baire Category Theorem.] Deduce that, for some , every can be written as with and
-
Paper 4, Section II, G
2014 commentDefine the spectrum and the approximate point spectrum of a bounded linear operator on a Banach space. Prove that and that is a closed and bounded subset of . [You may assume without proof that the set of invertible operators is open.]
Let be a hermitian operator on a non-zero Hilbert space. Prove that is not empty
Let be a non-empty, compact subset of . Show that there is a bounded linear operator with [You may assume without proof that a compact metric space is separable.]
-
Paper 4, Section II, I
2014 commentExplain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem.
We call a poset Bourbakian if every order-preserving map has a least fixed point . Suppose is Bourbakian, and let be order-preserving maps with for all ; show that . [Hint: Consider the function defined by if otherwise.]
Suppose is Bourbakian and is an order-preserving map from an ordinal to . Show that there is an order-preserving map whose fixed points are exactly the upper bounds of the set , and deduce that this set has a least upper bound.
Let be a chain with no greatest member. Using the Axiom of Choice and Hartogs' Lemma, show that there is an order-preserving map , for some ordinal , whose image has no upper bound in . Deduce that any Bourbakian poset is chain-complete.
-
Paper 3, Section II, I
2014 commentExplain what is meant by a structure for a first-order signature , and describe briefly how first-order terms and formulae in the language over are interpreted in a structure. Suppose that and are -structures, and that is a conjunction of atomic formulae over : show that an -tuple belongs to the interpretation of in if and only if and .
A first-order theory is called regular if its axioms all have the form
where and are (possibly empty) strings of variables and and are conjunctions of atomic formulae (possibly the empty conjunction ). Show that if and are models of a regular theory , then so is .
Now suppose that is a regular theory, and that a sentence of the form
is derivable from the axioms of , where and the are conjunctions of atomic formulae. Show that the sentence is derivable for some . [Hint: Suppose not, and use the Completeness Theorem to obtain a suitable family of -models .]
-
Paper 2, Section II, I
2014 commentWrite down the recursive definitions of ordinal addition, multiplication and exponentiation. Show that, for any nonzero ordinal , there exist unique ordinals and such that and .
Hence or otherwise show that (that is, the set of ordinals less than ) is closed under addition if and only if for some . Show also that an infinite ordinal is closed under multiplication if and only if for some .
[You may assume the standard laws of ordinal arithmetic, and the fact that for all .]
-
Paper 1, Section II, I
2014 commentExplain what is meant by saying that a binary relation is well-founded. Show that is well-founded if and only if, for any set and any function , there exists a unique function satisfying
for all . [Hint: For 'if', it suffices to take , with defined by
-
Paper 4, Section I, B
2014 commentThe concentration of a chemical in one dimension obeys the equations
State the physical interpretation of each equation.
Seek a similarity solution of the form , where . Find equations involving and from the differential equation and the integral. Show that these are satisfied by .
Find the solution for . Find and sketch the solution for .
-
Paper 3, Section , B
2014 commentAn epidemic model is given by
where are the susceptibles, are the infecteds, and and are positive parameters. The basic reproduction ratio is defined as , where is the total population size. Find a condition on for an epidemic to be possible if, initially, and is small but non-zero.
Now suppose a proportion of the population was vaccinated (with a completely effective vaccine) so that initially . On a sketch of the plane, mark the regions where an epidemic is still possible, where the vaccination will prevent an epidemic, and where no vaccination was necessary.
For the case when an epidemic is possible, show that , the proportion of the initially susceptible population that has not been infected by the end of an epidemic, satisfies
-
Paper 2, Section I, B
2014 commentConsider an experiment where two or three individuals are added to a population with probability and respectively per unit time. The death rate in the population is a constant per individual per unit time.
Write down the master equation for the probability that there are individuals in the population at time . From this, derive an equation for , where is the generating function
Find the solution for in steady state, and show that the mean and variance of the population size are given by
Hence show that, for a free choice of and subject to a given target mean, the experimenter can minimise the variance by only adding two individuals at a time.
-
Paper 1, Section I, B
2014 commentA population model for two species is given by
where and are positive parameters. Show that this may be rescaled to
and give and in terms of the original parameters.
For find all fixed points in , and analyse their stability. Assuming that both populations are present initially, what does this suggest will be the long-term outcome?
-
Paper 3, Section II, B
2014 commentA discrete-time model for breathing is given by
where is the volume of each breath in time step and is the concentration of carbon dioxide in the blood at the end of time step . The parameters and are all positive. Briefly explain the biological meaning of each of the above equations.
Find the steady state. For and determine the stability of the steady state.
For general (integer) , by seeking parameter values when the modulus of a perturbation to the steady state is constant, find the range of parameters where the solution is stable. What is the periodicity of the constant-modulus solution at the edge of this range? Comment on how the size of the range depends on .
This can be developed into a more realistic model by changing the term to in (2). Briefly explain the biological meaning of this change. Show that for both and the new steady state is stable if , where .
-
Paper 2, Section II, B
2014 commentAn activator-inhibitor system is described by the equations
where .
Find the range of for which the spatially homogeneous system has a stable equilibrium solution with and . Determine when the equilibrium is a stable focus, and sketch the phase diagram for this case (restricting attention to and .
For the case when the homogeneous system is stable, consider spatial perturbations proportional to of the solution found above. Briefly explain why the system will be stable to spatial perturbations with very small or very large . Find conditions for the system to be unstable to a spatial perturbation (for some range of which need not be given). Sketch the region satisfying these conditions in the plane.
Find , the critical wavenumber at the onset of instability, in terms of and .
-
Paper 4, Section II, F
2014 commentExplain what is meant by an integral basis for a number field. Splitting into the cases and , find an integral basis for where is a square-free integer. Justify your answer.
Find the fundamental unit in . Determine all integer solutions to the equation .
-
Paper 2, Section II, F
2014 comment(i) Show that each prime ideal in a number field divides a unique rational prime . Define the ramification index and residue class degree of such an ideal. State and prove a formula relating these numbers, for all prime ideals dividing a given rational prime , to the degree of over .
(ii) Show that if is a primitive th root of unity then . Deduce that if , where and are distinct primes, then is a unit in .
(iii) Show that if where is prime, then any prime ideal of dividing has ramification index at least . Deduce that .
-
Paper 1, Section II, F
2014 commentState a result involving the discriminant of a number field that implies that the class group is finite.
Use Dedekind's theorem to factor and 7 into prime ideals in . By factoring and , or otherwise, prove that the class group of is cyclic, and determine its order.
-
Paper 4, Section I, F
2014 commentState the Chinese Remainder Theorem.
Find all solutions to the simultaneous congruences
A positive integer is said to be square-free if it is the product of distinct primes. Show that there are 100 consecutive numbers that are not square-free.
-
Paper 3, Section I,
2014 commentShow that the continued fraction for is .
Hence, or otherwise, find positive integers and that satisfy the equation
Are there integers and such that ?
-
Paper 2, Section I, F
2014 commentShow that
Deduce that there are infinitely many primes.
-
Paper 1, Section I,
2014 commentDefine what it means for a number to be a pseudoprime to the base .
Show that if there is a base to which is not a pseudoprime, then is a pseudoprime to at most half of all possible bases.
Let be an integer greater than 1 such that is composite. Show that is a pseudoprime to the base 2 .
-
Paper 4, Section II, F
2014 commentDefine the Legendre and Jacobi symbols.
State the law of quadratic reciprocity for the Legendre symbol.
State the law of quadratic reciprocity for the Jacobi symbol, and deduce it from the corresponding result for the Legendre symbol.
Let be a prime with . Prove that the sum of the quadratic residues in the set is equal to the sum of the quadratic non-residues in this set.
For which primes is 7 a quadratic residue?
-
Paper 3, Section II, F
2014 commentState and prove Lagrange's theorem about polynomial congruences modulo a prime.
Define the Euler totient function .
Let be a prime and let be a positive divisor of . Show that there are exactly elements of with order
Deduce that is cyclic.
Let be a primitive root modulo . Show that must be a primitive root modulo .
Let be a primitive root modulo . Must it be a primitive root modulo ? Give a proof or a counterexample.
-
Paper 4, Section II, D
2014 commentLet be a real symmetric matrix with distinct real eigenvalues and a corresponding orthogonal basis of normalized real eigenvectors .
(i) Let satisfy . Given a unit vector , the iteration scheme
generates a sequence of vectors for . Assuming that with , prove that tends to as . What happens to if ? [Consider all cases.]
(ii) Describe how to implement an inverse-iteration algorithm to compute the eigenvalues and eigenvectors of , given some initial estimates for the eigenvalues.
(iii) Let . For iterates of an inverse-iteration algorithm with a fixed value of , show that if
where is small, then is of the same order of magnitude as .
(iv) Let still. Consider the iteration scheme
for , where , denotes the inner product. Show that with this scheme
-
Paper 2, Section II, D
2014 commentConsider the one-dimensional advection equation
subject to an initial condition . Consider discretization of this equation with finite differences on an equidistant space-time with step size in space and step size in time. Define the Courant number and explain briefly how such a discretization can be used to derive numerical schemes in which solutions and satisfy equations of the form
where the coefficients are independent of .
(i) Define the order of a numerical scheme such as (1). Define what a convergent numerical scheme is. Explain the notion of stability and state the Lax equivalence theorem that connects convergence and stability of numerical schemes for linear partial differential equations.
(ii) Consider the following example of (1):
Determine conditions on such that the scheme is stable and convergent. What is the order of this scheme?
-
Paper 3, Section II, D
2014 commentConsider the linear system
where and .
(i) Define the Jacobi iteration method with relaxation parameter for solving (1).
(ii) Assume that is a symmetric positive-definite matrix whose diagonal part is such that the matrix is also positive definite. Prove that the relaxed Jacobi iteration method always converges if the relaxation parameter is equal to
(iii) Let be the tridiagonal matrix with diagonal elements and off-diagonal elements , where . For which values of (expressed in terms of and ) does the relaxed Jacobi iteration method converge? What choice of gives the optimal convergence speed?
[You may quote without proof any relevant results about the convergence of iterative methods and about the eigenvalues of matrices.]
-
Paper 1, Section II, 40D
2014 comment(i) Consider the numerical approximation of the boundary-value problem
where are given constants and is a given smooth function on . A grid , on is given by
where and . Derive finite-difference approximations for , for , using at most one neighbouring grid point of on each side. Hence write down a numerical scheme to solve the problem, displaying explicitly the entries of the system matrix in the resulting system of linear equations , . What is the overall order of this numerical scheme? Explain briefly one strategy by which the order could be improved with the same grid.
(ii) Consider the numerical approximation of the boundary-value problem
where is an arbitrary, simply connected bounded domain with smooth boundary , and is a given smooth function. Define the 9-point formula used to approximate the Laplacian. Using this formula and an equidistant grid inside , define a numerical scheme for which the system matrix is symmetric and negative definite. Prove that the system matrix of your scheme has these properties for all choices of ordering of the grid points.
Part II, List of Questions
[TURN OVER
-
Paper 4, Section II, J
2014 commentA girl begins swimming from a point on the bank of a straight river. She swims at a constant speed relative to the water. The speed of the downstream current at a distance from the shore is . Hence her trajectory is described by
where is the angle at which she swims relative to the direction of the current.
She desires to reach a downstream point on the same bank as she starts, as quickly as possible. Construct the Hamiltonian for this problem, and describe how Pontryagin's maximum principle can be used to give necessary conditions that must hold on an optimal trajectory. Given that is positive, increasing and differentiable in , show that on an optimal trajectory
-
Paper 3, Section II, J
2014 commentA particle follows a discrete-time trajectory on given by
for , where is a fixed integer, is a real constant, is the position of the particle and is the control action at time , and is a sequence of independent random vectors with and .
Find the closed-loop control, i.e. the control action defined as a function of , that minimizes
where is given. [Note that this function is quadratic in , but linear in .]
Does the closed-loop control depend on or on ? Deduce the form of the optimal open-loop control.
-
Paper 2, Section II, J
2014 commentDescribe the elements of a discrete-time stochastic dynamic programming equation for the problem of maximizing the expected sum of non-negative rewards over an infinite horizon. Give an example to show that there may not exist an optimal policy. Prove that if a policy has a value function that satisfies the dynamic programming equation then the policy is optimal.
A squirrel collects nuts for the coming winter. There are plenty of nuts lying around, but each time the squirrel leaves its lair it risks being caught by a predator. Assume that the outcomes of the squirrel's journeys are independent, that it is caught with probability , and that it returns safely with a random weight of nuts, exponentially distributed with parameter . By solving the dynamic programming equation for the value function , find a policy maximizing the expected weight of nuts collected for the winter. Here the state variable takes values in (the weight of nuts so far collected) or (a no-return state when the squirrel is caught).
-
Paper 4, Section II, D
2014 comment(a) Derive the solution of the one-dimensional wave equation
with Cauchy data given by functions , and where and etc. Explain what is meant by the property of finite propagation speed for the wave equation. Verify that the solution to (1) satisfies this property.
(b) Consider the Cauchy problem
By considering the quantities
prove that solutions of (2) also satisfy the property of finite propagation speed.
(c) Define what is meant by a strongly continuous one-parameter group of unitary operators on a Hilbert space. Consider the Cauchy problem for the Schrödinger equation for :
[In the following you may use without proof the fact that there is an orthonormal set of (real-valued) Schwartz functions which are eigenfunctions of the differential operator with eigenvalues , i.e.
and which have the property that any function can be written uniquely as a sum which converges in the metric defined by the norm.]
Write down the solution to (3) in the case that is given by a finite sum and show that your formula extends to define a strongly continuous one-parameter group of unitary operators on the Hilbert space of square-integrable (complex-valued) functions, with inner product .
-
Paper 3, Section II, D
2014 comment(a) Consider variable-coefficient operators of the form
whose coefficients are defined on a bounded open set with smooth boundary . Let satisfy the condition of uniform ellipticity, namely
for suitably chosen positive numbers .
State and prove the weak maximum principle for solutions of . [Any results from linear algebra and calculus needed in your proof should be stated clearly, but need not be proved.]
(b) Consider the nonlinear elliptic equation
for satisfying the additional condition
Assume that . Prove that any two solutions of (1) which also satisfy (2) are equal.
Now let be a solution of and (2). Prove that if for all then for all . Prove that if then for all .
-
Paper 1, Section II, D
2014 commentState the Cauchy-Kovalevskaya theorem, including a definition of the term noncharacteristic.
For which values of the real number , and for which functions , does the CauchyKovalevskaya theorem ensure that the Cauchy problem
has a local solution?
Now consider the Cauchy problem (1) in the case that is a smooth -periodic function.
(i) Show that if there exists a unique smooth solution for all times, and show that for all there exists a number , independent of , such that
for all .
(ii) If does there exist a choice of for which (2) holds? Give a full justification for your answer.
-
Paper 2, Section II, D
2014 commentIn this question, functions are all real-valued, and
are the Sobolev spaces of functions -periodic in , for
State Parseval's theorem. For prove that the norm is equivalent to the norm || defined by
Consider the Cauchy problem
where is a smooth function which is -periodic in , and the initial value is also smooth and -periodic. Prove that if is a smooth solution which is -periodic in , then it satisfies
for some number which does not depend on or .
State the Lax-Milgram lemma. Prove, using the Lax-Milgram lemma, that if
with and , then there exists a weak solution to (1) of the form with . Does the same hold for all ? Briefly explain your answer.
-
Paper 4, Section II, A
2014 commentDefine the interaction picture for a quantum mechanical system with Schrödinger picture Hamiltonian and explain why the interaction and Schrödinger pictures give the same physical predictions for transition rates between eigenstates of . Derive the equation of motion for the interaction picture states .
A system consists of just two states and , with respect to which
Writing the interaction picture state as , show that the interaction picture equation of motion can be written as
where and . Hence show that satisfies
Given that , show that the solution takes the form
where is a frequency to be determined and is a complex constant of integration.
Substitute this solution for into to determine and, by imposing the normalization condition at , show that .
At time the system is in the state . Write down the probability of finding the system in the state at time .
-
Paper 3, Section II, A
2014 commentLet and denote the standard angular-momentum operators and states so that, in units where ,
Show that is unitary. Define
and
Find expressions for and as linear combinations of and . Briefly explain why represents a rotation of through angle about the 2-axis.
Show that
Express as a linear combination of the states . By expressing in terms of , use to determine the coefficients in this expansion.
A particle of spin 1 is in the state at time . It is subject to the Hamiltonian
where . At time the value of is measured and found to be . At time the value of is measured again and found to be . Show that the joint probability for these two values to be measured is
[The following result may be quoted:
-
Paper 2, Section II, A
2014 comment(i) Let and be the annihilation and creation operators, respectively, for a simple harmonic oscillator whose Hamiltonian is
with . Explain how the set of eigenstates of is obtained and deduce the corresponding eigenvalues. Show that
(ii) Consider a system whose unperturbed Hamiltonian is
where and all other commutators are zero. Find the degeneracies of the eigenvalues of with energies and .
The system is perturbed so that it is now described by the Hamiltonian
where . Using degenerate perturbation theory, calculate to the energies of the eigenstates associated with the level .
Write down the eigenstates, to , associated with these perturbed energies. By explicit evaluation show that they are in fact exact eigenstates of with these energies as eigenvalues.
-
Paper 1, Section II, A
2014 commentLet and be the position operator, momentum operator and Hamiltonian for a particle moving in one dimension. Let be the state vector for the particle. The position and momentum eigenstates have inner products
Show that
where and are the wavefunctions in the position representation and momentum representation, respectively. Show how and may be expressed in terms of each other.
For general , express in terms of , and hence write down the time-independent Schrödinger equation in the momentum representation satisfied by .
Consider now the case . Show that there is a bound state with energy , with wavefunction satisfying
Hence show that there is a unique value for and determine this value.
-
Paper 4, Section II, J
2014 commentSuppose you have at hand a pseudo-random number generator that can simulate an i.i.d. sequence of uniform distributed random variables for any . Construct an algorithm to simulate an i.i.d. sequence of standard normal random variables. [Should your algorithm depend on the inverse of any cumulative probability distribution function, you are required to provide an explicit expression for this inverse function.]
Suppose as a matter of urgency you need to approximately evaluate the integral
Find an approximation of this integral that requires simulation steps from your pseudo-random number generator, and which has stochastic accuracy
where Pr denotes the joint law of the simulated random variables. Justify your answer.
-
Paper 3, Section II, J
2014 commentState and prove Wilks' theorem about testing the simple hypothesis , against the alternative , in a one-dimensional regular parametric model . [You may use without proof the results from lectures on the consistency and asymptotic distribution of maximum likelihood estimators, as well as on uniform laws of large numbers. Necessary regularity conditions can be assumed without statement.]
Find the maximum likelihood estimator based on i.i.d. observations in a -model, . Deduce the limit distribution as of the sequence of statistics
where and are i.i.d. .
-
Paper 2, Section II, J
2014 commentIn a general decision problem, define the concepts of a Bayes rule and of admissibility. Show that a unique Bayes rule is admissible.
Consider i.i.d. observations from a , model. Can the maximum likelihood estimator of be a Bayes rule for estimating in quadratic risk for any prior distribution on that has a continuous probability density on Justify your answer.
Now model the as i.i.d. copies of , where is drawn from a prior that is a Gamma distribution with parameters and (given below). Show that the posterior distribution of is a Gamma distribution and find its parameters. Find the Bayes rule for estimating in quadratic risk for this prior. [The Gamma probability density function with parameters is given by
where is the usual Gamma function.]
Finally assume that the have actually been generated from a fixed Poisson distribution, where . Show that converges to zero in probability and deduce the asymptotic distribution of under the joint law of the random variables . [You may use standard results from lectures without proof provided they are clearly stated.]
-
Paper 1, Section II, J
2014 commentState without proof the inequality known as the Cramér-Rao lower bound in a parametric model . Give an example of a maximum likelihood estimator that attains this lower bound, and justify your answer.
Give an example of a parametric model where the maximum likelihood estimator based on observations is biased. State without proof an analogue of the Cramér-Rao inequality for biased estimators.
Define the concept of a minimax decision rule, and show that the maximum likelihood estimator based on in a model is minimax for estimating in quadratic risk.
-
Paper 4, Section II, K
2014 commentLet be a sequence of independent identically distributed random variables. Set .
(i) State the strong law of large numbers in terms of the random variables .
(ii) Assume now that the are non-negative and that their expectation is infinite. Let . What does the strong law of large numbers say about the limiting behaviour of , where ?
Deduce that almost surely.
Show that
Show that infinitely often almost surely.
(iii) Now drop the assumption that the are non-negative but continue to assume that . Show that, almost surely,
-
Paper 3, Section II, K
2014 comment(i) Let be a measure space. What does it mean to say that a function is a measure-preserving transformation?
What does it mean to say that is ergodic?
State Birkhoff's almost everywhere ergodic theorem.
(ii) Consider the set equipped with its Borel -algebra and Lebesgue measure. Fix an irrational number and define by
where addition in each coordinate is understood to be modulo 1 . Show that is a measurepreserving transformation. Is ergodic? Justify your answer.
Let be an integrable function on and let be the invariant function associated with by Birkhoff's theorem. Write down a formula for in terms of . [You are not expected to justify this answer.]
-
Paper 2, Section II,
2014 commentState and prove the monotone convergence theorem.
Let and be finite measure spaces. Define the product -algebra on .
Define the product measure on , and show carefully that is countably additive.
[You may use without proof any standard facts concerning measurability provided these are clearly stated.]
-
Paper 1, Section II,
2014 commentWhat is meant by the Borel -algebra on the real line ?
Define the Lebesgue measure of a Borel subset of using the concept of outer measure.
Let be the Lebesgue measure on . Show that, for any Borel set which is contained in the interval , and for any , there exist and disjoint intervals contained in such that, for , we have
where denotes the symmetric difference .
Show that there does not exist a Borel set contained in such that, for all intervals contained in ,
-
Paper 4, Section II, H
2014 commentLet .
(i) Sketch a proof that there is an isomorphism of topological groups
(ii) Let be the irreducible complex representation of of dimension 3. Compute the character of the (symmetric power) representation of for any . Show that the dimension of the space of invariants , meaning the subspace of where acts trivially, is 1 for even and 0 for odd. [Hint: You may find it helpful to restrict to the unit circle subgroup . The irreducible characters of may be quoted without proof.]
Using the fact that yields the standard 3-dimensional representation of , show that . Deduce that the ring of complex polynomials in three variables which are invariant under the action of is a polynomial ring in one generator. Find a generator for this polynomial ring.
-
Paper 3, Section II, H
2014 comment(i) State Frobenius' theorem for transitive permutation groups acting on a finite set. Define Frobenius group and show that any finite Frobenius group (with an appropriate action) satisfies the hypotheses of Frobenius' theorem.
(ii) Consider the group
where is prime, divides ( not necessarily prime), and has multiplicative order modulo (such elements exist since divides . Let be the subgroup of consisting of the powers of , so that . Write , and let be coset representatives for in .
(a) Show that has conjugacy classes and that a complete list of the classes comprises and .
(b) By observing that the derived subgroup , find 1-dimensional characters of . [Appropriate results may be quoted without proof.]
(c) Let . For denote by the character of defined by . By inducing these characters to , or otherwise, find distinct irreducible characters of degree .
-
Paper 2, Section II, H
2014 commentIn this question work over . Let be a subgroup of . State Mackey's restriction formula, defining all the terms you use. Deduce Mackey's irreducibility criterion.
Let (the dihedral group of order ) and let (the cyclic subgroup of of order ). Write down the inequivalent irreducible characters of . Determine the values of for which the induced character is irreducible.
-
Paper 1, Section II, H
2014 comment(i) Let be any field and let . Let be the Jordan block
Compute for each .
(ii) Let be a cyclic group of order , and let be an algebraically closed field of characteristic . Determine all the representations of on vector spaces over , up to equivalence. Which are irreducible? Which do not split as a direct sum , with and
-
Paper 3, Section II, H
2014 commentState the Uniformization Theorem.
Show that any domain of whose complement has more than one point is uniformized by the unit disc [You may use the fact that for the group of automorphisms consists of Möbius transformations, and for it consists of maps of the form with and .
Let be the torus , where is a lattice. Given , show that is uniformized by the unit .
Is it true that a holomorphic map from to a compact Riemann surface of genus two must be constant? Justify your answer.
-
Paper 2, Section II, H
2014 commentState and prove the Valency Theorem and define the degree of a non-constant holomorphic map between compact Riemann surfaces.
Let be a compact Riemann surface of genus and a holomorphic map of degree two. Find the cardinality of the set of ramification points of . Find also the cardinality of the set of branch points of . [You may use standard results from lectures provided they are clearly stated.]
Define as follows: if , then ; otherwise, where is the unique point such that and . Show that is a conformal equivalence with and id.
-
Paper 1, Section II, H
2014 commentIf is a Riemann surface and is a covering map of topological spaces, show that there is a conformal structure on such that is analytic.
Let be the complex polynomial . Consider the subspace of given by the equation , where denotes coordinates in , and let be the restriction of the projection map onto the first factor. Show that has the structure of a Riemann surface which makes an analytic map. If denotes projection onto the second factor, show that is also analytic. [You may assume that is connected.]
Find the ramification points and the branch points of both and . Compute also the ramification indices at the ramification points.
Assuming that it is possible to add a point to so that is a compact Riemann surface and extends to a holomorphic map such that , compute the genus of
-
Paper 4, Section I,
2014 commentConsider the normal linear model where the -vector of responses satisfies with and is an design matrix with full column rank. Write down a -level confidence set for .
Define the Cook's distance for the observation where is the th row of , and give its interpretation in terms of confidence sets for .
In the model above with and , you observe that one observation has Cook's distance 3.1. Would you be concerned about the influence of this observation? Justify your answer.
[Hint: You may find some of the following facts useful:
-
If , then .
-
If , then .
-
If , then
-
-
Paper 3, Section I,
2014 commentIn an experiment to study factors affecting the production of the plastic polyvinyl chloride , three experimenters each used eight devices to produce the PVC and measured the sizes of the particles produced. For each of the 24 combinations of device and experimenter, two size measurements were obtained.
The experimenters and devices used for each of the 48 measurements are stored in as factors in the objects experimenter and device respectively, with the measurements themselves stored in the vector psize. The following analysis was performed in .

Let and denote the design matrices obtained by model.matrix(fit) and model.matrix (fit0) respectively, and let denote the response psize. Let and denote orthogonal projections onto the column spaces of and respectively.
For each of the following quantities, write down their numerical values if they appear in the analysis of variance table above; otherwise write 'unknown'.
Out of the two models that have been fitted, which appears to be the more appropriate for the data according to the analysis performed, and why?
-
-
Paper 2, Section I, K
2014 commentDefine the concept of an exponential dispersion family. Show that the family of scaled binomial distributions , with and , is of exponential dispersion family form.
Deduce the mean of the scaled binomial distribution from the exponential dispersion family form.
What is the canonical link function in this case?
-
Paper 1, Section , K
2014 commentWrite down the model being fitted by the following command, where and is an matrix with real-valued entries.
fit poisson)
Write down the log-likelihood for the model. Explain why the command
predict (fit, type "response"
gives the answer 0, by arguing based on the log-likelihood you have written down. [Hint: Recall that if then
for .]
-
Paper 4, Section II, K
2014 commentIn a study on infant respiratory disease, data are collected on a sample of 2074 infants. The information collected includes whether or not each infant developed a respiratory disease in the first year of their life; the gender of each infant; and details on how they were fed as one of three categories (breast-fed, bottle-fed and supplement). The data are tabulated in as follows:
Write down the model being fit by the commands on the following page:

The following (slightly abbreviated) output from is obtained.

Briefly explain the justification for the standard errors presented in the output above.
Explain the relevance of the output of the following code to the data being studied, justifying your answer:
[1]
[Hint: It may help to recall that if then
Let be the deviance of the model fitted by the following command.
fit glm (disease/total gender + food + gender:food,
family = binomial, weights = total
What is the numerical value of ? Which of the two models that have been fitted should you prefer, and why?
-
Paper 1, Section II,
2014 commentConsider the normal linear model where the -vector of responses satisfies with . Here is an matrix of predictors with full column rank where , and is an unknown vector of regression coefficients. Let be the matrix formed from the first columns of , and partition as where and . Denote the orthogonal projections onto the column spaces of and by and respectively.
It is desired to test the null hypothesis against the alternative hypothesis . Recall that the -test for testing against rejects for large values of
Show that , and hence prove that the numerator and denominator of are independent under either hypothesis.
Show that
where .
[In this question you may use the following facts without proof: is an orthogonal projection with rank ; any orthogonal projection matrix satisfies , where and if then when
-
Paper 4, Section II, E
2014 commentThe Dieterici equation of state of a gas is
where is the pressure, is the volume divided by the number of particles, is the temperature, and is the Boltzmann constant. Provide a physical interpretation for the constants and .
Briefly explain how the Dieterici equation captures the liquid-gas phase transition. What is the maximum temperature at which such a phase transition can occur?
The Gibbs free energy is given by
where is the energy and is the entropy. Explain why the Gibbs free energy is proportional to the number of particles in the system.
On either side of a first-order phase transition the Gibbs free energies are equal. Use this fact to derive the Clausius-Clapeyron equation for a line along which there is a first-order liquid-gas phase transition,
where is the latent heat which you should define.
Assume that the volume of liquid is negligible compared to the volume of gas and that the latent heat is constant. Further assume that the gas can be well approximated by the ideal gas law. Solve to obtain an equation for the phase-transition line in the plane.
-
Paper 3, Section II, E
2014 commentIn the grand canonical ensemble, at temperature and chemical potential , what is the probability of finding a system in a state with energy and particle number ?
A particle with spin degeneracy and mass moves in spatial dimensions with dispersion relation . Compute the density of states . [You may denote the area of a unit -dimensional sphere as .]
Treating the particles as non-interacting fermions, determine the energy of a gas in terms of the pressure and volume .
Derive an expression for the Fermi energy in terms of the number density of particles. Compute the degeneracy pressure at zero temperature in terms of the number of particles and the Fermi energy.
Show that at high temperatures the gas obeys the ideal gas law (up to small corrections which you need not compute).
-
Paper 2, Section II, E
2014 commentBriefly describe the microcanonical, canonical and grand canonical ensembles. Why do they agree in the thermodynamic limit?
A harmonic oscillator in one spatial dimension has Hamiltonian
Here and are the momentum and position of the oscillator, is its mass and its frequency. The harmonic oscillator is placed in contact with a heat bath at temperature . What is the relevant ensemble?
Treating the harmonic oscillator classically, compute the mean energy , the energy fluctuation and the heat capacity .
Treating the harmonic oscillator quantum mechanically, compute the mean energy , the energy fluctuation and the heat capacity .
In what limit of temperature do the classical and quantum results agree? Explain why they differ away from this limit. Describe an experiment for which this difference has implications.
-
Paper 1, Section II, E
2014 commentWrite down the equation of state and the internal energy of a monatomic ideal gas.
Describe the meaning of an adiabatic process. Derive the equation for an adiabatic process in the pressure-volume plane for a monatomic ideal gas.
Briefly describe the Carnot cycle. Sketch the Carnot cycle in the plane and in the temperature entropy plane.
The Diesel cycle is an idealised version of the process realised in the Diesel engine. It consists of the following four reversible steps:

Sketch the Diesel cycle for a monatomic gas in the plane and the plane. Determine the equations for the curves and in the plane.
The efficiency of the cycle is defined as
where is the heat entering the gas in step and is the heat leaving the gas in step . Calculate as a function of the temperatures at points and .
-
Paper 4, Section II, K
2014 commentWrite down the Black-Scholes partial differential equation (PDE), and explain briefly its relevance to option pricing.
Show how a change of variables reduces the Black-Scholes PDE to the heat equation:
where is a given boundary function.
Consider the following numerical scheme for solving the heat equation on the equally spaced grid where and and , and . We approximate by where
and is a constant and the operator is the matrix with non-zero entries and . By considering what happens when , show that the finite-difference scheme is stable if and only if
where . For what values of is the scheme unconditionally stable?
-
Paper 3, Section II,
2014 commentDerive the Black-Scholes formula for the time- 0 price of a European call option with expiry and strike written on an asset with volatility and time-0 price , and where is the riskless rate of interest. Explain what is meant by the delta hedge for this option, and determine it explicitly.
In terms of the Black-Scholes call option price formula , find the time- 0 price of a forward-starting option, which pays at time , where and are given. Find the price of an option which pays at time . How would this option be hedged?
-
Paper 1, Section II, K
2014 commentSuppose that denotes the vector of prices of assets at times , and that denotes the vector of the numbers of the different assets held by an investor from time to time . Assuming that asset 0 is a bank account paying zero interest, that is, for all , explain what is meant by the statement that the portfolio process is self-financing. If the portfolio process is self-financing, prove that for any
where , and .
Suppose now that the are independent with common distribution. Let
where and the infimum is taken over all self-financing portfolio processes with . Explain why should satisfy the equation
If is a positive-definite symmetric matrix satisfying the equation
show that has a solution of the form .
-
Paper 2, Section II, K
2014 commentAn agent has expected-utility preferences over his possible wealth at time 1 ; that is, the wealth is preferred to wealth if and only if , where the function is strictly concave and twice continuously differentiable. The agent can trade in a market, with the time-1 value of his portfolio lying in an affine space of random variables. Assuming cash can be held between time 0 and time 1 , define the agent's time-0 utility indifference price for a contingent claim with time-1 value . Assuming any regularity conditions you may require, prove that the map is concave.

Consider a market with two claims with time-1 values and . Their joint distribution is
At time 0 , arbitrary quantities of the claim can be bought at price , but is not marketed. Derive an explicit expression for in the case where
where is a given constant.
-
Paper 4, Section I,
2014 commentState Liouville's theorem on approximation of algebraic numbers by rationals.
Prove that the number is transcendental.
-
Paper 3, Section I, G
2014 commentState Runge's theorem about uniform approximation of holomorphic functions by polynomials.
Let be the subset of non-negative real numbers and let
Prove that there is a sequence of complex polynomials which converges to the function uniformly on each compact subset of .
-
Paper 2, Section I, G
2014 commentState Chebyshev's equal ripple criterion.
Let
Show that if where are real constants with , then
-
Paper 1, Section I,
2014 comment(i) State Brouwer's fixed point theorem in the plane and an equivalent theorem concerning mapping a triangle to its boundary .
(ii) Let be a matrix with positive real entries. Use the theorems you stated in (i) to prove that has an eigenvector with positive entries.
-
Paper 2, Section II, G
2014 commentLet be a continuous map never taking the value 0 and satisfying . Define the degree (or winding number) of about 0 . Prove the following:
(i) , where .
(ii) If is continuous, and for each , then .
(iii) If , are continuous maps with , which converge to uniformly on as , then for sufficiently large .
Let be a continuous map such that and for each . Deduce from the results of (ii) and (iii) that .
[You may not use homotopy invariance of the winding number without proof.]
-
Paper 3, Section II, G
2014 commentDefine what is meant by a nowhere dense set in a metric space. State a version of the Baire Category Theorem. Show that any complete non-empty metric space without isolated points is uncountable.
Let be the set of real numbers whose decimal expansion does not use the digit 6 . (A terminating decimal representation is used when it exists.) Show that there exists a real number which cannot be written as with and .
-
Paper 4, Section II, C
2014 commentA one-dimensional shock wave propagates at a constant speed along a tube aligned with the -axis and containing a perfect gas. In the reference frame where the shock is at rest at , the gas has speed , density and pressure in the region and speed , density and pressure in the region .
Write down equations of conservation of mass, momentum and energy across the shock. Show that
where is the ratio of specific heats.
From now on, assume and let . Show that .
The increase in entropy from to is given by , where is a positive constant. Show that is a monotonic function of .
If , deduce that and , where and are the sound speeds in and , respectively. Given that must have the same sign as and , interpret these inequalities physically in terms of the properties of the flow upstream and downstream of the shock.
-
Paper 2, Section II, 38C
2014 commentThe function satisfies the equation
Derive the dispersion relation, and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. In the case of a localised initial disturbance, will it be the shortest or the longest waves that are to be found at the front of a dispersing wave packet? Do the wave crests move faster or slower than the wave packet?
Give the solution to the initial-value problem for which at
and is real. Use the method of stationary phase to obtain an approximation for for fixed and large . If, in addition, , deduce an approximation for the sequence of times at which .
You are given that decreases like for large . Give a brief physical explanation why this rate of decay is slower than for . What can be said about for large if ? [Detailed calculation is not required in these cases.]
[You may assume that for
-
Paper 3, Section II, 39C
2014 commentThe equations describing small-amplitude motions in a stably stratified, incompressible, inviscid fluid are
where is the background stratification, and are the perturbations about an undisturbed hydrostatic state, is the velocity, and .
Show that
stating any approximation made, and define the Brunt-Väisälä frequency .
Deduce the dispersion relation for plane harmonic waves with wavevector . Calculate the group velocity and verify that it is perpendicular to .
Such a stably stratified fluid with a uniform value of occupies the region above a moving lower boundary . Find the velocity field generated by the boundary motion for the case , where and is a constant.
For the case , sketch the orientation of the wave crests, the direction of propagation of the crests, and the direction of the group velocity.
-
Paper 1, Section II, C
2014 commentState the equations that relate strain to displacement and stress to strain in a uniform, linear, isotropic elastic solid with Lamé moduli and . In the absence of body forces, the Cauchy momentum equation for the infinitesimal displacements is
where is the density and the stress tensor. Show that both the dilatation and the rotation satisfy wave equations, and find the wave-speeds and .
A plane harmonic -wave with wavevector lying in the plane is incident from at an oblique angle on the planar interface between two elastic solids with different densities and elastic moduli. Show in a diagram the directions of all the reflected and transmitted waves, labelled with their polarisations, assuming that none of these waves are evanescent. State the boundary conditions on components of and that would, in principle, determine the amplitudes.
Now consider a plane harmonic P-wave of unit amplitude incident with on the interface between two elastic (and inviscid) liquids with wave-speed and modulus in and wave-speed and modulus in . Obtain solutions for the reflected and transmitted waves. Show that the amplitude of the reflected wave is zero if
where and
-
Paper 2, Section II, C
2014 commentWrite down the Euler-Lagrange equation for the integral
An ant is walking on the surface of a sphere, which is parameterised by angle from top of sphere) and ) (azimuthal angle). The sphere is sticky towards the top and the bottom and so the ant's speed is proportional to . Show that the ant's fastest route between two points will be of the form
for some constants and . need not be determined.]
-
Paper 4, Section II, C
2014 commentConsider the integral
Show that if satisfies the Euler-Lagrange equation, then
An axisymmetric soap film is formed between two circular wires at . The wires both have radius . Show that the shape that minimises the surface area takes the form
Show that there exist two possible that satisfy the boundary conditions for sufficiently large.
Show that for these solutions the second variation is given by
where is an axisymmetric perturbation with .
-
Paper 1, Section II, C
2014 commentLet and be complex matrices.
(i) The commutator of and is defined to be
Show that and for . Show further that the trace of vanishes.
(ii) A skew-Hermitian matrix is one which satisfies , where denotes the Hermitian conjugate. Show that if and are skew-Hermitian then so is .
(iii) Let be the linear map from to the set of complex matrices given by
where
Prove that for any is traceless and skew-Hermitian. By considering pairs such as , or otherwise, show that for ,
(iv) Using the result of part (iii), or otherwise, prove that if is a traceless skewHermitian matrix then there exist matrices such that . [You may use geometrical properties of vectors in without proof.]